![Logarithm plots](https://upload.wikimedia.org/wikipedia/commons/8/81/Logarithm_plots.png)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the logarithm is the
inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X\t ...
to
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
. That means the logarithm of a number to the base is the
exponent
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of is , or . The logarithm of to ''base'' is denoted as , or without parentheses, , or even without the explicit base, , when no confusion is possible, or when the base does not matter such as in
big O notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
.
The logarithm base is called the decimal or
common logarithm
In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered i ...
and is commonly used in science and engineering. The
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
has the number
as its base; its use is widespread in mathematics and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, because of its very simple
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
. The
binary logarithm
In mathematics, the binary logarithm () is the power to which the number must be raised to obtain the value . That is, for any real number ,
:x=\log_2 n \quad\Longleftrightarrow\quad 2^x=n.
For example, the binary logarithm of is , the b ...
uses base and is frequently used in
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
.
Logarithms were introduced by
John Napier
John Napier of Merchiston (; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8th Laird of Merchiston. His Latinized name was Ioann ...
in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors and others to perform high-accuracy computations more easily. Using
logarithm tables, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because the logarithm of a
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
is the
sum of the logarithms of the factors:
:
provided that , and are all positive and . The
slide rule
The slide rule is a mechanical analog computer which is used primarily for multiplication and division, and for functions such as exponents, roots, logarithms, and trigonometry. It is not typically designed for addition or subtraction, which is ...
, also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes from
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
, who connected them to the
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
in the 18th century, and who also introduced the letter as the base of natural logarithms.
Logarithmic scale
A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
s reduce wide-ranging quantities to smaller scopes. For example, the
decibel
The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a po ...
(dB) is a
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
used to express
ratio as logarithms, mostly for signal power and amplitude (of which
sound pressure
Sound pressure or acoustic pressure is the local pressure deviation from the ambient (average or equilibrium) atmospheric pressure, caused by a sound wave. In air, sound pressure can be measured using a microphone, and in water with a hydrophone ...
is a common example). In chemistry,
pH is a logarithmic measure for the
acid
In computer science, ACID ( atomicity, consistency, isolation, durability) is a set of properties of database transactions intended to guarantee data validity despite errors, power failures, and other mishaps. In the context of databases, a sequ ...
ity of an
aqueous solution
An aqueous solution is a solution in which the solvent is water. It is mostly shown in chemical equations by appending (aq) to the relevant chemical formula. For example, a solution of table salt, or sodium chloride (NaCl), in water would be re ...
. Logarithms are commonplace in scientific
formula
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...
e, and in measurements of the
complexity of algorithms and of geometric objects called
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
s. They help to describe
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
ratios of
musical intervals, appear in formulas counting
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s or
approximating factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \t ...
s, inform some models in
psychophysics
Psychophysics quantitatively investigates the relationship between physical stimuli and the sensations and perceptions they produce. Psychophysics has been described as "the scientific study of the relation between stimulus and sensation" or, m ...
, and can aid in
forensic accounting
Forensic accounting, forensic accountancy or financial forensics is the specialty practice area of accounting that investigates whether firms engage in financial reporting misconduct. Forensic accountants apply a range of skills and methods to d ...
.
The concept of logarithm as the inverse of exponentiation extends to other mathematical structures as well. However, in general settings, the logarithm tends to be a multi-valued function. For example, the
complex logarithm
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
is the multi-valued
inverse of the complex exponential function. Similarly, the
discrete logarithm
In mathematics, for given real numbers ''a'' and ''b'', the logarithm log''b'' ''a'' is a number ''x'' such that . Analogously, in any group ''G'', powers ''b'k'' can be defined for all integers ''k'', and the discrete logarithm log''b' ...
is the multi-valued inverse of the exponential function in finite groups; it has uses in
public-key cryptography
Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic alg ...
.
Motivation
Addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
,
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
, and
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
are three of the most fundamental arithmetic operations. The inverse of addition is subtraction, and the inverse of multiplication is
division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
*Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting ...
. Similarly, a logarithm is the inverse operation of
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
. Exponentiation is when a number , the ''base'', is raised to a certain power , the ''exponent'', to give a value ; this is denoted
:
For example, raising to the power of gives :
The logarithm of base is the inverse operation, that provides the output from the input . That is,
is equivalent to
if is a positive
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
. (If is not a positive real number, both exponentiation and logarithm can be defined but may take several values, which makes definitions much more complicated.)
One of the main historical motivations of introducing logarithms is the formula
:
which allowed (before the invention of computers) reducing computation of multiplications and divisions to additions, subtractions and
logarithm table
In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered i ...
looking.
Definition
Given a positive
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
such that , the ''logarithm'' of a positive real number with respect to base is the exponent by which must be raised to yield . In other words, the logarithm of to base is the unique real number such that
.
The logarithm is denoted "" (pronounced as "the logarithm of to base ", "the logarithm of ", or most commonly "the log, base , of ").
An equivalent and more succinct definition is that the function is the
inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X\t ...
to the function
.
Examples
* , since .
* Logarithms can also be negative:
since
* is approximately 2.176, which lies between 2 and 3, just as 150 lies between and .
* For any base , and , since and , respectively.
Logarithmic identities
Several important formulas, sometimes called ''logarithmic identities'' or ''logarithmic laws'', relate logarithms to one another.
Product, quotient, power, and root
The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. The logarithm of the -th power of a number is '' ''times the logarithm of the number itself; the logarithm of a -th root is the logarithm of the number divided by . The following table lists these identities with examples. Each of the identities can be derived after substitution of the logarithm definitions
or
in the left hand sides.
Change of base
The logarithm can be computed from the logarithms of and with respect to an arbitrary base using the following formula:
:
Starting from the defining identity
:
we can apply to both sides of this equation, to get
:
.
Solving for
yields:
:
,
showing the conversion factor from given
-values to their corresponding
-values to be
Typical
scientific calculators
A scientific calculator is an electronic calculator, either desktop or handheld, designed to perform mathematical operations. They have completely replaced slide rules and are used in both educational and professional settings.
In some areas ...
calculate the logarithms to bases 10 and . Logarithms with respect to any base can be determined using either of these two logarithms by the previous formula:
:
Given a number and its logarithm to an unknown base , the base is given by:
:
which can be seen from taking the defining equation
to the power of
Particular bases
![Log4](https://upload.wikimedia.org/wikipedia/commons/a/ae/Log4.svg)
Among all choices for the base, three are particularly common. These are , (the
irrational
Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
mathematical constant ≈ 2.71828), and (the
binary logarithm
In mathematics, the binary logarithm () is the power to which the number must be raised to obtain the value . That is, for any real number ,
:x=\log_2 n \quad\Longleftrightarrow\quad 2^x=n.
For example, the binary logarithm of is , the b ...
). In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, the logarithm base is widespread because of analytical properties explained below. On the other hand, logarithms are easy to use for manual calculations in the
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
number system:
:
Thus, is related to the number of
decimal digit
A numerical digit (often shortened to just digit) is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent numbers in a positional numeral system. The name "digit" comes from the fact that the ten digits (Latin ...
s of a positive integer : the number of digits is the smallest
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
strictly bigger than . For example, is approximately 3.15. The next integer is 4, which is the number of digits of 1430. Both the natural logarithm and the logarithm to base two are used in
information theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...
, corresponding to the use of
nat
Nat or NAT may refer to:
Computing
* Network address translation (NAT), in computer networking
Organizations
* National Actors Theatre, New York City, U.S.
* National AIDS trust, a British charity
* National Archives of Thailand
* National As ...
s or
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
s as the fundamental units of information, respectively. Binary logarithms are also used in
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, where the
binary system
A binary system is a system of two astronomical bodies which are close enough that their gravitational attraction causes them to orbit each other around a barycenter ''(also see animated examples)''. More restrictive definitions require that thi ...
is ubiquitous; in
music theory
Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the "rudiments", that are needed to understand music notation (ke ...
, where a pitch ratio of two (the
octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
) is ubiquitous and the number of
cents between any two pitches is the binary logarithm, times 1200, of their ratio (that is, 100 cents per
equal-temperament semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
); and in
photography
Photography is the art, application, and practice of creating durable images by recording light, either electronically by means of an image sensor, or chemically by means of a light-sensitive material such as photographic film. It is employed ...
to measure
exposure value
In photography, exposure value (EV) is a number that represents a combination of a camera's shutter speed and f-number, such that all combinations that yield the same exposure have the same EV (for any fixed scene luminance). Exposure value is a ...
s,
light levels,
exposure time
In photography, shutter speed or exposure time is the length of time that the film or digital sensor inside the camera is exposed to light (that is, when the camera's shutter is open) when taking a photograph.
The amount of light that re ...
s,
aperture
In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture and focal length of an optical system determine the cone angle of a bundle of rays that come to a focus in the image plane.
An opt ...
s, and
film speed
Film speed is the measure of a photographic film's sensitivity to light, determined by sensitometry and measured on various numerical scales, the most recent being the ISO system. A closely related ISO system is used to describe the relations ...
s in "stops".
The following table lists common notations for logarithms to these bases and the fields where they are used. Many disciplines write instead of , when the intended base can be determined from the context. The notation also occurs. The "ISO notation" column lists designations suggested by the
International Organization for Standardization
The International Organization for Standardization (ISO ) is an international standard development organization composed of representatives from the national standards organizations of member countries. Membership requirements are given in Ar ...
(
ISO 80000-2
ISO 80000 or IEC 80000 is an international standard introducing the International System of Quantities (ISQ).
It was developed and promulgated jointly by the International Organization for Standardization (ISO) and the International Electrote ...
). Because the notation has been used for all three bases (or when the base is indeterminate or immaterial), the intended base must often be inferred based on context or discipline. In computer science, usually refers to , and in mathematics usually refers to . In other contexts, often means .
History
The history of logarithms in seventeenth-century Europe is the discovery of a new
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
that extended the realm of analysis beyond the scope of algebraic methods. The method of logarithms was publicly propounded by
John Napier
John Napier of Merchiston (; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8th Laird of Merchiston. His Latinized name was Ioann ...
in 1614, in a book titled ''
Mirifici Logarithmorum Canonis Descriptio
''Mirifici Logarithmorum Canonis Descriptio'' (Description of the Wonderful Canon of Logarithms, 1614) and ''Mirifici Logarithmorum Canonis Constructio'' (Construction of the Wonderful Canon of Logarithms, 1619) are two books in Latin by John N ...
'' (''Description of the Wonderful Rule of Logarithms''). Prior to Napier's invention, there had been other techniques of similar scopes, such as the
prosthaphaeresis
Prosthaphaeresis (from the Greek ''προσθαφαίρεσις'') was an algorithm used in the late 16th century and early 17th century for approximate multiplication and division using formulas from trigonometry. For the 25 years preceding the ...
or the use of tables of progressions, extensively developed by
Jost Bürgi
Jost Bürgi (also ''Joost, Jobst''; Latinisation of names, Latinized surname ''Burgius'' or ''Byrgius''; 28 February 1552 – 31 January 1632), active primarily at the courts in Kassel and Prague, was a Swiss clockmaker, a maker of astronomica ...
around 1600.
Napier coined the term for logarithm in Middle Latin, “logarithmus,” derived from the Greek, literally meaning, “ratio-number,” from ''logos'' “proportion, ratio, word” + ''arithmos'' “number”.
The
common logarithm
In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered i ...
of a number is the index of that power of ten which equals the number. Speaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
as the “order of a number”. The first real logarithms were heuristic methods to turn multiplication into addition, thus facilitating rapid computation. Some of these methods used tables derived from trigonometric identities. Such methods are called
prosthaphaeresis
Prosthaphaeresis (from the Greek ''προσθαφαίρεσις'') was an algorithm used in the late 16th century and early 17th century for approximate multiplication and division using formulas from trigonometry. For the 25 years preceding the ...
.
Invention of the
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
now known as the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
began as an attempt to perform a
quadrature of a rectangular
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
by
Grégoire de Saint-Vincent
Grégoire de Saint-Vincent - in latin : Gregorius a Sancto Vincentio, in dutch : Gregorius van St-Vincent - (8 September 1584 Bruges – 5 June 1667 Ghent) was a Flemish Jesuit and mathematician. He is remembered for his work on quadrature of th ...
, a Belgian Jesuit residing in Prague. Archimedes had written ''
The Quadrature of the Parabola
''Quadrature of the Parabola'' ( el, Τετραγωνισμὸς παραβολῆς) is a treatise on geometry, written by Archimedes in the 3rd century BC and addressed to his Alexandrian acquaintance Dositheus. It contains 24 propositions rega ...
'' in the third century BC, but a quadrature for the hyperbola eluded all efforts until Saint-Vincent published his results in 1647. The relation that the logarithm provides between a
geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For e ...
in its
argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
and an
arithmetic progression
An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
of values, prompted
A. A. de Sarasa
Alphonse Antonio de Sarasa was a Jesuit mathematician who contributed to the understanding of logarithms, particularly as areas under a hyperbola.
Alphonse de Sarasa was born in 1618, in Nieuwpoort (Belgium), Nieuwpoort in Flanders. In 1632 he ...
to make the connection of Saint-Vincent's quadrature and the tradition of logarithms in
prosthaphaeresis
Prosthaphaeresis (from the Greek ''προσθαφαίρεσις'') was an algorithm used in the late 16th century and early 17th century for approximate multiplication and division using formulas from trigonometry. For the 25 years preceding the ...
, leading to the term “hyperbolic logarithm”, a synonym for natural logarithm. Soon the new function was appreciated by
Christiaan Huygens
Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists of ...
, and
James Gregory. The notation Log y was adopted by
Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
in 1675, and the next year he connected it to the
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
Before Euler developed his modern conception of complex natural logarithms,
Roger Cotes
Roger Cotes (10 July 1682 – 5 June 1716) was an English mathematician, known for working closely with Isaac Newton by proofreading the second edition of his famous book, the '' Principia'', before publication. He also invented the quadratur ...
had a nearly equivalent result when he showed in 1714 that
:
.
Logarithm tables, slide rules, and historical applications
By simplifying difficult calculations before calculators and computers became available, logarithms contributed to the advance of science, especially
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
. They were critical to advances in
surveying
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ca ...
,
celestial navigation
Celestial navigation, also known as astronavigation, is the practice of position fixing using stars and other celestial bodies that enables a navigator to accurately determine their actual current physical position in space (or on the surface of ...
, and other domains.
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
called logarithms
::"...
admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations."
As the function is the inverse function of , it has been called an antilogarithm. Nowadays, this function is more commonly called an
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
.
Log tables
A key tool that enabled the practical use of logarithms was the ''
table of logarithms''. The first such table was compiled by
Henry Briggs Henry Briggs may refer to:
*Henry Briggs (mathematician) (1561–1630), English mathematician
*Henry Perronet Briggs (1793–1844), English painter
*Henry George Briggs (1824–1872), English merchant, traveller, and orientalist
*Henry Shaw Briggs ...
in 1617, immediately after Napier's invention but with the innovation of using 10 as the base. Briggs' first table contained the
common logarithm
In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered i ...
s of all integers in the range from 1 to 1000, with a precision of 14 digits. Subsequently, tables with increasing scope were written. These tables listed the values of for any number in a certain range, at a certain precision. Base-10 logarithms were universally used for computation, hence the name common logarithm, since numbers that differ by factors of 10 have logarithms that differ by integers. The common logarithm of can be separated into an
integer part
In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least inte ...
and a
fractional part
The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part can ...
, known as the characteristic and
mantissa. Tables of logarithms need only include the mantissa, as the characteristic can be easily determined by counting digits from the decimal point. The characteristic of is one plus the characteristic of , and their mantissas are the same. Thus using a three-digit log table, the logarithm of 3542 is approximated by
:
Greater accuracy can be obtained by
interpolation
In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.
In engineering and science, one often has a n ...
:
:
The value of can be determined by reverse look up in the same table, since the logarithm is a
monotonic function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
.
Computations
The product and quotient of two positive numbers and ' were routinely calculated as the sum and difference of their logarithms. The product or quotient came from looking up the antilogarithm of the sum or difference, via the same table:
:
and
:
For manual calculations that demand any appreciable precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such as
prosthaphaeresis
Prosthaphaeresis (from the Greek ''προσθαφαίρεσις'') was an algorithm used in the late 16th century and early 17th century for approximate multiplication and division using formulas from trigonometry. For the 25 years preceding the ...
, which relies on
trigonometric identities
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
.
Calculations of powers and
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
are reduced to multiplications or divisions and lookups by
:
and
:
Trigonometric calculations were facilitated by tables that contained the common logarithms of
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
s.
Slide rules
Another critical application was the
slide rule
The slide rule is a mechanical analog computer which is used primarily for multiplication and division, and for functions such as exponents, roots, logarithms, and trigonometry. It is not typically designed for addition or subtraction, which is ...
, a pair of logarithmically divided scales used for calculation. The non-sliding logarithmic scale,
Gunter's rule
Edmund Gunter (158110 December 1626), was an English clergyman, mathematician, geometer and astronomer of Welsh descent. He is best remembered for his mathematical contributions which include the invention of the Gunter's chain, the Gunter's q ...
, was invented shortly after Napier's invention.
William Oughtred
William Oughtred ( ; 5 March 1574 – 30 June 1660), also Owtred, Uhtred, etc., was an Kingdom of England, English mathematician and Anglican ministry, Anglican clergyman.'Oughtred (William)', in P. Bayle, translated and revised by J.P. Bernar ...
enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other. Numbers are placed on sliding scales at distances proportional to the differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms, as illustrated here:
For example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part. The slide rule was an essential calculating tool for engineers and scientists until the 1970s, because it allows, at the expense of precision, much faster computation than techniques based on tables.
Analytic properties
A deeper study of logarithms requires the concept of a ''
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
''. A function is a rule that, given one number, produces another number. An example is the function producing the -th power of from any real number , where the base is a fixed number. This function is written as . When is positive and unequal to 1, we show below that is invertible when considered as a function from the reals to the positive reals.
Existence
Let be a positive real number not equal to 1 and let .
It is a standard result in real analysis that any continuous strictly monotonic function is bijective between its domain and range. This fact follows from the
intermediate value theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval.
This has two import ...
.
[, section III.3] Now, is
strictly increasing (for ), or strictly decreasing (for ),
is continuous, has domain
, and has range
. Therefore, is a bijection from
to
. In other words, for each positive real number , there is exactly one real number such that
.
We let
denote the inverse of . That is, is the unique real number such that
. This function is called the base- ''logarithm function'' or ''logarithmic function'' (or just ''logarithm'').
Characterization by the product formula
The function can also be essentially characterized by the product formula
:
More precisely, the logarithm to any base is the only
increasing function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
''f'' from the positive reals to the reals satisfying and
:
Graph of the logarithm function
![Logarithm inversefunctiontoexp](https://upload.wikimedia.org/wikipedia/commons/4/49/Logarithm_inversefunctiontoexp.svg)
As discussed above, the function is the inverse to the exponential function
. Therefore, Their
graphs
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
correspond to each other upon exchanging the - and the -coordinates (or upon reflection at the diagonal line ), as shown at the right: a point on the graph of yields a point on the graph of the logarithm and vice versa. As a consequence,
diverges to infinity (gets bigger than any given number) if grows to infinity, provided that is greater than one. In that case, is an
increasing function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
. For , tends to minus infinity instead. When approaches zero, goes to minus infinity for (plus infinity for , respectively).
Derivative and antiderivative
![Logarithm derivative](https://upload.wikimedia.org/wikipedia/commons/5/57/Logarithm_derivative.svg)
Analytic properties of functions pass to their inverses.
Thus, as is a continuous and
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
, so is . Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of evaluates to by the properties of the
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
, the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
implies that the derivative of is given by
:
That is, the
slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
of the
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
touching the graph of the logarithm at the point equals .
The derivative of is ; this implies that is the unique
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
of that has the value 0 for . It is this very simple formula that motivated to qualify as "natural" the natural logarithm; this is also one of the main reasons of the importance of the
constant .
The derivative with a generalized functional argument is
:
The quotient at the right hand side is called the
logarithmic derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function ''f'' is defined by the formula
\frac
where f' is the derivative of ''f''. Intuitively, this is the infinitesimal relative change in ''f''; ...
of '. Computing by means of the derivative of is known as
logarithmic differentiation
In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function ''f'',
:(\ln f)' = \frac \quad \implies \quad f' = f \cdot (\ ...
. The antiderivative of the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
is:
:
Related formulas, such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.
Integral representation of the natural logarithm
![Natural logarithm integral](https://upload.wikimedia.org/wikipedia/commons/d/df/Natural_logarithm_integral.svg)
The
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
of can be defined as the
definite integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...
:
:
This definition has the advantage that it does not rely on the exponential function or any trigonometric functions; the definition is in terms of an integral of a simple reciprocal. As an integral, equals the area between the -axis and the graph of the function , ranging from to . This is a consequence of the
fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
and the fact that the derivative of is . Product and power logarithm formulas can be derived from this definition. For example, the product formula is deduced as:
:
The equality (1) splits the integral into two parts, while the equality (2) is a change of variable (). In the illustration below, the splitting corresponds to dividing the area into the yellow and blue parts. Rescaling the left hand blue area vertically by the factor and shrinking it by the same factor horizontally does not change its size. Moving it appropriately, the area fits the graph of the function again. Therefore, the left hand blue area, which is the integral of from to is the same as the integral from 1 to . This justifies the equality (2) with a more geometric proof.
![Natural logarithm product formula proven geometrically](https://upload.wikimedia.org/wikipedia/commons/9/9b/Natural_logarithm_product_formula_proven_geometrically.svg)
The power formula may be derived in a similar way:
:
The second equality uses a change of variables (
integration by substitution
In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can ...
), .
The sum over the reciprocals of natural numbers,
:
is called the
harmonic series. It is closely tied to the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
: as tends to
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
, the difference,
:
converges (i.e. gets arbitrarily close) to a number known as the
Euler–Mascheroni constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural l ...
. This relation aids in analyzing the performance of algorithms such as
quicksort
Quicksort is an efficient, general-purpose sorting algorithm. Quicksort was developed by British computer scientist Tony Hoare in 1959 and published in 1961, it is still a commonly used algorithm for sorting. Overall, it is slightly faster than ...
.
Transcendence of the logarithm
Real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s that are not
algebraic are called
transcendental; for example,
and ''
e'' are such numbers, but
is not.
Almost all
In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the math ...
real numbers are transcendental. The logarithm is an example of a
transcendental function
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function.
In other words, a transcendental function "transcends" algebra in that it cannot be expressed alge ...
. The
Gelfond–Schneider theorem
In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers.
History
It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider.
Statement
: If ''a'' and ''b'' are ...
asserts that logarithms usually take transcendental, i.e. "difficult" values.
Calculation
![Logarithm keys](https://upload.wikimedia.org/wikipedia/commons/8/88/Logarithm_keys.jpg)
Logarithms are easy to compute in some cases, such as . In general, logarithms can be calculated using
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
or the
arithmetic–geometric mean, or be retrieved from a precalculated
logarithm table
In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered i ...
that provides a fixed precision.
Newton's method
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...
, an iterative method to solve equations approximately, can also be used to calculate the logarithm, because its inverse function, the exponential function, can be computed efficiently. Using look-up tables,
CORDIC
CORDIC (for "coordinate rotation digital computer"), also known as Volder's algorithm, or: Digit-by-digit method Circular CORDIC (Jack E. Volder), Linear CORDIC, Hyperbolic CORDIC (John Stephen Walther), and Generalized Hyperbolic CORDIC (GH C ...
-like methods can be used to compute logarithms by using only the operations of addition and
bit shifts. Moreover, the
binary logarithm algorithm calculates
recursively
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
, based on repeated squarings of , taking advantage of the relation
:
Power series
Taylor series
![Taylor approximation of natural logarithm](https://upload.wikimedia.org/wikipedia/commons/0/02/Taylor_approximation_of_natural_logarithm.gif)
For any real number that satisfies , the following formula holds:
:
This is a shorthand for saying that can be approximated to a more and more accurate value by the following expressions:
:
For example, with the third approximation yields 0.4167, which is about 0.011 greater than . This
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
approximates with arbitrary precision, provided the number of summands is large enough. In elementary calculus, is therefore the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
of this series. It is the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
of the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
at . The Taylor series of provides a particularly useful approximation to when is small, , since then
:
For example, with the first-order approximation gives , which is less than 5% off the correct value 0.0953.
Although the sequence for
only converges for
, a neat trick can fix this.
:
As
for all
, the sequence converges for the same range of .
Inverse hyperbolic tangent
Another series is based on the
inverse hyperbolic tangent
In mathematics, the inverse hyperbolic functions are the inverse functions of the hyperbolic functions.
For a given value of a hyperbolic function, the corresponding inverse hyperbolic function provides the corresponding hyperbolic angle. The ...
function:
:
for any real number .
Using
sigma notation
In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matr ...
, this is also written as
:
This series can be derived from the above Taylor series. It converges quicker than the Taylor series, especially if is close to 1. For example, for , the first three terms of the second series approximate with an error of about . The quick convergence for close to 1 can be taken advantage of in the following way: given a low-accuracy approximation and putting
:
the logarithm of is:
:
The better the initial approximation is, the closer is to 1, so its logarithm can be calculated efficiently. can be calculated using the
exponential series, which converges quickly provided is not too large. Calculating the logarithm of larger can be reduced to smaller values of by writing , so that .
A closely related method can be used to compute the logarithm of integers. Putting
in the above series, it follows that:
:
If the logarithm of a large integer is known, then this series yields a fast converging series for , with a
rate of convergence
In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence (x_n) that converges to x^* is said to have ''order of co ...
of
.
Arithmetic–geometric mean approximation
The
arithmetic–geometric mean yields high precision approximations of the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
. Sasaki and Kanada showed in 1982 that it was particularly fast for precisions between 400 and 1000 decimal places, while Taylor series methods were typically faster when less precision was needed. In their work is approximated to a precision of (or precise bits) by the following formula (due to
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
):
:
Here denotes the
arithmetic–geometric mean of and . It is obtained by repeatedly calculating the average (
arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...
) and
(
geometric mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
) of and then let those two numbers become the next and . The two numbers quickly converge to a common limit which is the value of . is chosen such that
:
to ensure the required precision. A larger makes the calculation take more steps (the initial and are farther apart so it takes more steps to converge) but gives more precision. The constants and can be calculated with quickly converging series.
Feynman's algorithm
While at
Los Alamos National Laboratory
Los Alamos National Laboratory (often shortened as Los Alamos and LANL) is one of the sixteen research and development laboratories of the United States Department of Energy (DOE), located a short distance northwest of Santa Fe, New Mexico, ...
working on the
Manhattan Project
The Manhattan Project was a research and development undertaking during World War II that produced the first nuclear weapons. It was led by the United States with the support of the United Kingdom and Canada. From 1942 to 1946, the project w ...
,
Richard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superflu ...
developed a bit-processing algorithm, to compute the logarithm, that is similar to long division and was later used in the
Connection Machine
A Connection Machine (CM) is a member of a series of massively parallel supercomputers that grew out of doctoral research on alternatives to the traditional von Neumann architecture of computers by Danny Hillis at Massachusetts Institute of Techno ...
. The algorithm uses the fact that every real number is representable as a product of distinct factors of the form . The algorithm sequentially builds that product , starting with and : if , then it changes to . It then increases
by one regardless. The algorithm stops when is large enough to give the desired accuracy. Because is the sum of the terms of the form corresponding to those for which the factor was included in the product , may be computed by simple addition, using a table of for all . Any base may be used for the logarithm table.
Applications
![NautilusCutawayLogarithmicSpiral](https://upload.wikimedia.org/wikipedia/commons/0/08/NautilusCutawayLogarithmicSpiral.jpg)
Logarithms have many applications inside and outside mathematics. Some of these occurrences are related to the notion of
scale invariance
In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality.
The technical term ...
. For example, each chamber of the shell of a
nautilus
The nautilus (, ) is a pelagic marine mollusc of the cephalopod family Nautilidae. The nautilus is the sole extant family of the superfamily Nautilaceae and of its smaller but near equal suborder, Nautilina.
It comprises six living species in t ...
is an approximate copy of the next one, scaled by a constant factor. This gives rise to a
logarithmic spiral
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). Mor ...
.
Benford's law
Benford's law, also known as the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation that in many real-life sets of numerical data, the leading digit is likely to be small.Arno Berger and Theodore ...
on the distribution of leading digits can also be explained by scale invariance. Logarithms are also linked to
self-similarity
__NOTOC__
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
. For example, logarithms appear in the analysis of algorithms that solve a problem by dividing it into two similar smaller problems and patching their solutions. The dimensions of self-similar geometric shapes, that is, shapes whose parts resemble the overall picture are also based on logarithms.
Logarithmic scale
A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
s are useful for quantifying the relative change of a value as opposed to its absolute difference. Moreover, because the logarithmic function grows very slowly for large , logarithmic scales are used to compress large-scale scientific data. Logarithms also occur in numerous scientific formulas, such as the
Tsiolkovsky rocket equation
Konstantin Eduardovich Tsiolkovsky (russian: Константи́н Эдуа́рдович Циолко́вский , , p=kənstɐnʲˈtʲin ɪdʊˈardəvʲɪtɕ tsɨɐlˈkofskʲɪj , a=Ru-Konstantin Tsiolkovsky.oga; – 19 September 1935) ...
, the
Fenske equation
The Fenske equation in continuous fractional distillation is an equation used for calculating the minimum number of theoretical plates required for the separation of a binary feed stream by a fractionation column that is being operated at total ...
, or the
Nernst equation
In electrochemistry, the Nernst equation is a Thermodynamics#Chemical thermodynamics, chemical thermodynamical relationship that permits the calculation of the reduction potential of a reaction (half-cell or electrochemical cell, full cell reacti ...
.
Logarithmic scale
![Germany Hyperinflation](https://upload.wikimedia.org/wikipedia/commons/4/4f/Germany_Hyperinflation.svg)
Scientific quantities are often expressed as logarithms of other quantities, using a ''logarithmic scale''. For example, the
decibel
The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a po ...
is a
unit of measurement
A unit of measurement is a definite magnitude (mathematics), magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can ...
associated with
logarithmic-scale
A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
quantities
Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit ...
. It is based on the common logarithm of
ratio
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
s—10 times the common logarithm of a
power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may a ...
ratio or 20 times the common logarithm of a
voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
ratio. It is used to quantify the loss of voltage levels in transmitting electrical signals, to describe power levels of sounds in
acoustics
Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
, and the
absorbance
Absorbance is defined as "the logarithm of the ratio of incident to transmitted radiant power through a sample (excluding the effects on cell walls)". Alternatively, for samples which scatter light, absorbance may be defined as "the negative lo ...
of light in the fields of
spectrometry and
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
. The
signal-to-noise ratio
Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to the noise power, often expressed in deci ...
describing the amount of unwanted
noise
Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arise ...
in relation to a (meaningful)
signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...
is also measured in decibels. In a similar vein, the
peak signal-to-noise ratio
Peak signal-to-noise ratio (PSNR) is an engineering term for the ratio between the maximum possible power of a signal and the power of corrupting noise that affects the fidelity of its representation. Because many signals have a very wide dynamic ...
is commonly used to assess the quality of sound and
image compression
Image compression is a type of data compression applied to digital images, to reduce their cost for storage or transmission. Algorithms may take advantage of visual perception and the statistical properties of image data to provide superior r ...
methods using the logarithm.
The strength of an earthquake is measured by taking the common logarithm of the energy emitted at the quake. This is used in the
moment magnitude scale
The moment magnitude scale (MMS; denoted explicitly with or Mw, and generally implied with use of a single M for magnitude) is a measure of an earthquake's magnitude ("size" or strength) based on its seismic moment. It was defined in a 1979 pape ...
or the
Richter magnitude scale
The Richter scale —also called the Richter magnitude scale, Richter's magnitude scale, and the Gutenberg–Richter scale—is a measure of the strength of earthquakes, developed by Charles Francis Richter and presented in his landmark 1935 ...
. For example, a 5.0 earthquake releases 32 times and a 6.0 releases 1000 times the energy of a 4.0.
Apparent magnitude
Apparent magnitude () is a measure of the brightness of a star or other astronomical object observed from Earth. An object's apparent magnitude depends on its intrinsic luminosity, its distance from Earth, and any extinction of the object's li ...
measures the brightness of stars logarithmically. In
chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
the negative of the decimal logarithm, the decimal , is indicated by the letter p.
For instance,
pH is the decimal cologarithm of the
activity of
hydronium
In chemistry, hydronium (hydroxonium in traditional British English) is the common name for the aqueous cation , the type of oxonium ion produced by protonation of water. It is often viewed as the positive ion present when an Arrhenius acid is d ...
ions (the form
hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic, an ...
ion
An ion () is an atom or molecule with a net electrical charge.
The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conven ...
s take in water). The activity of hydronium ions in neutral water is 10
−7 mol·L−1, hence a pH of 7. Vinegar typically has a pH of about 3. The difference of 4 corresponds to a ratio of 10
4 of the activity, that is, vinegar's hydronium ion activity is about .
Semilog (log–linear) graphs use the logarithmic scale concept for visualization: one axis, typically the vertical one, is scaled logarithmically. For example, the chart at the right compresses the steep increase from 1 million to 1 trillion to the same space (on the vertical axis) as the increase from 1 to 1 million. In such graphs,
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
s of the form appear as straight lines with
slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
equal to the logarithm of .
Log-log graphs scale both axes logarithmically, which causes functions of the form to be depicted as straight lines with slope equal to the exponent . This is applied in visualizing and analyzing
power law
In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a proportional relative change in the other quantity, inde ...
s.
Psychology
Logarithms occur in several laws describing
human perception
Perception () is the organization, identification, and interpretation of sensory information in order to represent and understand the presented information or environment. All perception involves signals that go through the nervous system ...
:
Hick's law proposes a logarithmic relation between the time individuals take to choose an alternative and the number of choices they have.
Fitts's law
Fitts's law (often cited as Fitts' law) is a predictive model of human movement primarily used in human–computer interaction and ergonomics. The law predicts that the time required to rapidly move to a target area is a function of the ratio betw ...
predicts that the time required to rapidly move to a target area is a logarithmic function of the distance to and the size of the target. In
psychophysics
Psychophysics quantitatively investigates the relationship between physical stimuli and the sensations and perceptions they produce. Psychophysics has been described as "the scientific study of the relation between stimulus and sensation" or, m ...
, the
Weber–Fechner law
The Weber–Fechner laws are two related hypotheses in the field of psychophysics, known as Weber's law and Fechner's law. Both laws relate to human perception, more specifically the relation between the actual change in a physical stimulus and ...
proposes a logarithmic relationship between
stimulus
A stimulus is something that causes a physiological response. It may refer to:
*Stimulation
**Stimulus (physiology), something external that influences an activity
**Stimulus (psychology), a concept in behaviorism and perception
*Stimulus (economi ...
and
sensation
Sensation (psychology) refers to the processing of the senses by the sensory system.
Sensation or sensations may also refer to:
In arts and entertainment In literature
*Sensation (fiction), a fiction writing mode
*Sensation novel, a British ...
such as the actual vs. the perceived weight of an item a person is carrying. (This "law", however, is less realistic than more recent models, such as
Stevens's power law
Stevens' power law is an empirical relationship in psychophysics between an increased intensity or strength in a physical stimulus and the perceived magnitude increase in the sensation created by the stimulus. It is often considered to supersede ...
.)
Psychological studies found that individuals with little mathematics education tend to estimate quantities logarithmically, that is, they position a number on an unmarked line according to its logarithm, so that 10 is positioned as close to 100 as 100 is to 1000. Increasing education shifts this to a linear estimate (positioning 1000 10 times as far away) in some circumstances, while logarithms are used when the numbers to be plotted are difficult to plot linearly.
Probability theory and statistics
![PDF-log normal distributions](https://upload.wikimedia.org/wikipedia/commons/a/ae/PDF-log_normal_distributions.svg)
Logarithms arise in
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
: the
law of large numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...
dictates that, for a
fair coin
In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin. In th ...
, as the number of coin-tosses increases to infinity, the observed proportion of heads
approaches one-half. The fluctuations of this proportion about one-half are described by the
law of the iterated logarithm
In probability theory, the law of the iterated logarithm describes the magnitude of the fluctuations of a random walk. The original statement of the law of the iterated logarithm is due to A. Ya. Khinchin (1924). Another statement was given by A ...
.
Logarithms also occur in
log-normal distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
s. When the logarithm of a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
has a
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
, the variable is said to have a log-normal distribution. Log-normal distributions are encountered in many fields, wherever a variable is formed as the product of many independent positive random variables, for example in the study of turbulence.
Logarithms are used for
maximum-likelihood estimation
In statistics, maximum likelihood estimation (MLE) is a method of estimation theory, estimating the Statistical parameter, parameters of an assumed probability distribution, given some observed data. This is achieved by Mathematical optimization, ...
of parametric
statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repres ...
s. For such a model, the
likelihood function
The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood funct ...
depends on at least one
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
that must be estimated. A maximum of the likelihood function occurs at the same parameter-value as a maximum of the logarithm of the likelihood (the "''log likelihood''"), because the logarithm is an increasing function. The log-likelihood is easier to maximize, especially for the multiplied likelihoods for
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
random variables.
Benford's law
Benford's law, also known as the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation that in many real-life sets of numerical data, the leading digit is likely to be small.Arno Berger and Theodore ...
describes the occurrence of digits in many
data set A data set (or dataset) is a collection of data. In the case of tabular data, a data set corresponds to one or more database tables, where every column of a table represents a particular variable, and each row corresponds to a given record of the ...
s, such as heights of buildings. According to Benford's law, the probability that the first decimal-digit of an item in the data sample is (from 1 to 9) equals , ''regardless'' of the unit of measurement. Thus, about 30% of the data can be expected to have 1 as first digit, 18% start with 2, etc. Auditors examine deviations from Benford's law to detect fraudulent accounting.
The
logarithm transformation
In statistics, data transformation is the application of a deterministic mathematical function to each point in a data set—that is, each data point ''zi'' is replaced with the transformed value ''yi'' = ''f''(''zi''), where ''f'' is a functio ...
is a type of
data transformation
In computing, data transformation is the process of converting data from one format or structure into another format or structure. It is a fundamental aspect of most data integrationCIO.com. Agile Comes to Data Integration. Retrieved from: http ...
used to bring the empirical distribution closer to the assumed one.
Computational complexity
Analysis of algorithms
In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a function that re ...
is a branch of
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
that studies the
performance
A performance is an act of staging or presenting a play, concert, or other form of entertainment. It is also defined as the action or process of carrying out or accomplishing an action, task, or function.
Management science
In the work place ...
of
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
s (computer programs solving a certain problem).
[, pp. 1–2] Logarithms are valuable for describing algorithms that
divide a problem into smaller ones, and join the solutions of the subproblems.
For example, to find a number in a sorted list, the
binary search algorithm
In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the m ...
checks the middle entry and proceeds with the half before or after the middle entry if the number is still not found. This algorithm requires, on average, comparisons, where is the list's length. Similarly, the
merge sort
In computer science, merge sort (also commonly spelled as mergesort) is an efficient, general-purpose, and comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the order of equal elements is the same i ...
algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results. Merge sort algorithms typically require a time
approximately proportional to . The base of the logarithm is not specified here, because the result only changes by a constant factor when another base is used. A constant factor is usually disregarded in the analysis of algorithms under the standard
uniform cost model
In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a Function (mathem ...
.
A function is said to
grow logarithmically if is (exactly or approximately) proportional to the logarithm of . (Biological descriptions of organism growth, however, use this term for an exponential function.) For example, any
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
can be represented in
binary form
Binary form is a musical form in 2 related sections, both of which are usually repeated. Binary is also a structure used to choreograph dance. In music this is usually performed as A-A-B-B.
Binary form was popular during the Baroque period, of ...
in no more than
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
s. In other words, the amount of
memory
Memory is the faculty of the mind by which data or information is encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If past events could not be remembered, ...
needed to store grows logarithmically with .
Entropy and chaos
Entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
is broadly a measure of the disorder of some system. In
statistical thermodynamics
In physics, statistical mechanics is a mathematical framework that applies Statistics, statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the ma ...
, the entropy ''S'' of some physical system is defined as
:
The sum is over all possible states of the system in question, such as the positions of gas particles in a container. Moreover, is the probability that the state is attained and is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
. Similarly,
entropy in information theory measures the quantity of information. If a message recipient may expect any one of possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as bits.
Lyapunov exponent
In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with ini ...
s use logarithms to gauge the degree of chaoticity of a
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
. For example, for a particle moving on an oval billiard table, even small changes of the initial conditions result in very different paths of the particle. Such systems are
chaotic
Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kid ...
in a
deterministic
Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
way, because small measurement errors of the initial state predictably lead to largely different final states. At least one Lyapunov exponent of a deterministically chaotic system is positive.
Fractals
![Sierpinski dimension](https://upload.wikimedia.org/wikipedia/commons/3/37/Sierpinski_dimension.svg)
Logarithms occur in definitions of the
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
s. Fractals are geometric objects that are self-similar in the sense that small parts reproduce, at least roughly, the entire global structure. The
Sierpinski triangle (pictured) can be covered by three copies of itself, each having sides half the original length. This makes the
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a ...
of this structure . Another logarithm-based notion of dimension is obtained by
counting the number of boxes needed to cover the fractal in question.
Music
Logarithms are related to musical tones and
intervals
Interval may refer to:
Mathematics and physics
* Interval (mathematics), a range of numbers
** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets
* A statistical level of measurement
* Interval e ...
. In
equal temperament
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...
, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or
pitch, of the individual tones. For example, the
note ''A'' has a frequency of 440
Hz and
''B-flat'' has a frequency of 466 Hz. The interval between ''A'' and ''B-flat'' is a
semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
, as is the one between ''B-flat'' and
''B'' (frequency 493 Hz). Accordingly, the frequency ratios agree:
:
Therefore, logarithms can be used to describe the intervals: an interval is measured in semitones by taking the logarithm of the
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
ratio, while the logarithm of the frequency ratio expresses the interval in
cents, hundredths of a semitone. The latter is used for finer encoding, as it is needed for non-equal temperaments.
Number theory
Natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
s are closely linked to
counting prime numbers (2, 3, 5, 7, 11, ...), an important topic in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
. For any
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, the quantity of
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s less than or equal to is denoted . The
prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
asserts that is approximately given by
:
in the sense that the ratio of and that fraction approaches 1 when tends to infinity. As a consequence, the probability that a randomly chosen number between 1 and is prime is inversely
proportional to the number of decimal digits of . A far better estimate of is given by the
offset logarithmic integral
In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is ...
function , defined by
:
The
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
, one of the oldest open mathematical
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
s, can be stated in terms of comparing and . The
Erdős–Kac theorem In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely ...
describing the number of distinct
prime factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s also involves the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
.
The logarithm of ''n''
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \t ...
, , is given by
:
This can be used to obtain
Stirling's formula
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less ...
, an approximation of for large .
Generalizations
Complex logarithm
![Complex number illustration multiple arguments](https://upload.wikimedia.org/wikipedia/commons/8/83/Complex_number_illustration_multiple_arguments.svg)
All the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s that solve the equation
:
are called ''complex logarithms'' of , when is (considered as) a complex number. A complex number is commonly represented as , where and are real numbers and is an
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, the square of which is −1. Such a number can be visualized by a point in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, as shown at the right. The
polar form
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
encodes a non-zero complex number by its
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
, that is, the (positive, real) distance to the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
, and an angle between the real () axis'' '' and the line passing through both the origin and . This angle is called the
argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
of .
The absolute value of is given by
:
Using the geometrical interpretation of
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
and
cosine and their periodicity in , any complex number may be denoted as
:
for any integer number . Evidently the argument of is not uniquely specified: both and are valid arguments of for all integers , because adding
radians
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
or ''k''⋅360° to corresponds to "winding" around the origin counter-clock-wise by
turns. The resulting complex number is always , as illustrated at the right for . One may select exactly one of the possible arguments of as the so-called ''principal argument'', denoted , with a capital , by requiring to belong to one, conveniently selected turn, e.g. or . These regions, where the argument of is uniquely determined are called
''branches'' of the argument function.
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for an ...
connects the
trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
and
cosine to the
complex exponential
The exponential function is a mathematical Function (mathematics), function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponentiation, exponent). Unless otherwise specified, the term generally refers to the positiv ...
:
:
Using this formula, and again the periodicity, the following identities hold:
:
where is the unique real natural logarithm, denote the complex logarithms of , and is an arbitrary integer. Therefore, the complex logarithms of , which are all those complex values for which the power of equals , are the infinitely many values
:
for arbitrary integers .
Taking such that is within the defined interval for the principal arguments, then is called the ''principal value'' of the logarithm, denoted , again with a capital . The principal argument of any positive real number is 0; hence is a real number and equals the real (natural) logarithm. However, the above formulas for logarithms of products and powers
do ''not'' generalize to the principal value of the complex logarithm.
The illustration at the right depicts , confining the arguments of to the interval . This way the corresponding branch of the complex logarithm has discontinuities all along the negative real axis, which can be seen in the jump in the hue there. This discontinuity arises from jumping to the other boundary in the same branch, when crossing a boundary, i.e. not changing to the corresponding -value of the continuously neighboring branch. Such a locus is called a
branch cut
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, a ...
. Dropping the range restrictions on the argument makes the relations "argument of ", and consequently the "logarithm of ",
multi-valued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...
s.
Inverses of other exponential functions
Exponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm. For example, the
logarithm of a matrix
In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exp ...
is the (multi-valued) inverse function of the
matrix exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives ...
. Another example is the
''p''-adic logarithm, the inverse function of the
''p''-adic exponential. Both are defined via Taylor series analogous to the real case. In the context of
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, the
exponential map maps the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
at a point of a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
to a
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of that point. Its inverse is also called the logarithmic (or log) map.
In the context of
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
s exponentiation is given by repeatedly multiplying one group element with itself. The
discrete logarithm
In mathematics, for given real numbers ''a'' and ''b'', the logarithm log''b'' ''a'' is a number ''x'' such that . Analogously, in any group ''G'', powers ''b'k'' can be defined for all integers ''k'', and the discrete logarithm log''b' ...
is the integer ' solving the equation
:
where is an element of the group. Carrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups. This asymmetry has important applications in
public key cryptography
Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic alg ...
, such as for example in the
Diffie–Hellman key exchange
Diffie–Hellman key exchangeSynonyms of Diffie–Hellman key exchange include:
* Diffie–Hellman–Merkle key exchange
* Diffie–Hellman key agreement
* Diffie–Hellman key establishment
* Diffie–Hellman key negotiation
* Exponential key exc ...
, a routine that allows secure exchanges of
cryptographic
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or '' -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adve ...
keys over unsecured information channels.
Zech's logarithm
Zech logarithms are used to implement addition in finite fields when elements are represented as powers of a generator \alpha.
Zech logarithms are named after Julius Zech, and are also called Jacobi logarithms, after Carl G. J. Jacobi who used t ...
is related to the discrete logarithm in the multiplicative group of non-zero elements of a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
.
Further logarithm-like inverse functions include the ''double logarithm'' , the ''
super- or hyper-4-logarithm'' (a slight variation of which is called
iterated logarithm
In computer science, the iterated logarithm of n, written n (usually read "log star"), is the number of times the logarithm function must be iteratively applied before the result is less than or equal to 1. The simplest formal definition i ...
in computer science), the
Lambert W function
In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the Branch point, branches of the converse relation of the function , where is any complex number and is the expone ...
, and the
logit
In statistics, the logit ( ) function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially in data transformations.
Mathematically, the logit is the ...
. They are the inverse functions of the
double exponential function
A double exponential function is a constant raised to the power of an exponential function. The general formula is f(x) = a^=a^ (where ''a''>1 and ''b''>1), which grows much more quickly than an exponential function. For example, if ''a'' = ''b ...
,
tetration
In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common.
Under the definition as rep ...
, of , and of the
logistic function
A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with equation
f(x) = \frac,
where
For values of x in the domain of real numbers from -\infty to +\infty, the S-curve shown on the right is obtained, with the ...
, respectively.
Related concepts
From the perspective of
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, the identity expresses a group isomorphism between positive real number, reals under multiplication and reals under addition. Logarithmic functions are the only continuous isomorphisms between these groups. By means of that isomorphism, the Haar measure (Lebesgue measure) on the reals corresponds to the Haar measure on the positive reals. The non-negative reals not only have a multiplication, but also have addition, and form a semiring, called the probability semiring; this is in fact a semifield. The logarithm then takes multiplication to addition (log multiplication), and takes addition to log addition (LogSumExp), giving an isomorphism of semirings between the probability semiring and the log semiring.
logarithmic form, Logarithmic one-forms appear in complex analysis and algebraic geometry as differential forms with logarithmic Pole (complex analysis), poles.
The polylogarithm is the function defined by
:
It is related to the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
by . Moreover, equals the Riemann zeta function .
See also
* Decimal exponent (dex)
* Exponential function
* Index of logarithm articles
Notes
References
External links
*
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*
*
Khan Academy: Logarithms, free online micro lectures*
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Logarithms,
Elementary special functions
Scottish inventions
Additive functions