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The natural logarithm of a number is its
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
to the base of the
mathematical constant A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...
, which is an
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 ...
and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the base is implicit, simply .
Parentheses A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
are sometimes added for clarity, giving , , or . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity. The natural logarithm of is the
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 ...
to which would have to be raised to equal . For example, is , because . The natural logarithm of itself, , is , because , while the natural logarithm of is , since . The natural logarithm can be defined for any 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 ...
as the
area under the curve 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 ...
from to (with the area being negative when ). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural". The definition of the natural logarithm can then be extended to give logarithm values for negative numbers and for all non-zero
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, although this leads to a
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 ...
: see
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 ...
for more. The natural logarithm function, if considered as a
real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real fun ...
of a positive real variable, 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 ...
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 ...
, leading to the identities: :\begin e^ &= x \qquad \text x \text \\ \ln e^x &= x \qquad \text x \text \end Like all logarithms, the natural logarithm maps multiplication of positive numbers into addition: : \ln( x \cdot y ) = \ln x + \ln y~. Logarithms can be defined for any positive base other than 1, not only . However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and can be defined in terms of the latter, \log_bx = \ln x / \ln b = \ln x \cdot \log_b e. Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity. For example, logarithms are used to solve for the
half-life Half-life (symbol ) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable ato ...
, decay constant, or unknown time in
exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
problems. They are important in many branches of mathematics and scientific disciplines, and are used to solve problems involving compound interest.


History

The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and
Alphonse Antonio 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 in Flanders. In 1632 he was admitted as a no ...
before 1649. Their work involved quadrature of the
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 ...
with equation , by determination of the area of
hyperbolic sector A hyperbolic sector is a region of the Cartesian plane bounded by a hyperbola and two rays from the origin to it. For example, the two points and on the rectangular hyperbola , or the corresponding region when this hyperbola is re-scaled and i ...
s. Their solution generated the requisite "hyperbolic logarithm"
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 ...
, which had the properties now associated with the natural logarithm. An early mention of the natural logarithm was by
Nicholas Mercator Nicholas (Nikolaus) Mercator (c. 1620, Holstein – 1687, Versailles), also known by his German name Kauffmann, was a 17th-century mathematician. He was born in Eutin, Schleswig-Holstein, Germany and educated at Rostock and Leyden after which he ...
in his work ''Logarithmotechnia'', published in 1668, although the mathematics teacher
John Speidell John Speidell ( fl. 1600–1634) was an English mathematician. He is known for his early work on the calculation of logarithms. Speidell was a mathematics teacher in London and one of the early followers of the work John Napier had previously done ...
had already compiled a table of what in fact were effectively natural logarithms in 1619. It has been said that Speidell's logarithms were to the base , but this is not entirely true due to complications with the values being expressed as integers.


Notational conventions

The notations and both refer unambiguously to the natural logarithm of , and without an explicit base may also refer to the natural logarithm. This usage is common in mathematics, along with some scientific contexts as well as in many
programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming ...
s.Including C,
C++ C++ (pronounced "C plus plus") is a high-level general-purpose programming language created by Danish computer scientist Bjarne Stroustrup as an extension of the C programming language, or "C with Classes". The language has expanded significan ...
,
SAS SAS or Sas may refer to: Arts, entertainment, and media * ''SAS'' (novel series), a French book series by Gérard de Villiers * ''Shimmer and Shine'', an American animated children's television series * Southern All Stars, a Japanese rock ba ...
,
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation ...
,
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...
, Fortran, and some
BASIC BASIC (Beginners' All-purpose Symbolic Instruction Code) is a family of general-purpose, high-level programming languages designed for ease of use. The original version was created by John G. Kemeny and Thomas E. Kurtz at Dartmouth College ...
dialects
In some other contexts such as
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 ...
, however, can be used to denote the common (base 10) logarithm. It may also refer to the binary (base 2) logarithm in the context 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 ...
, particularly in the context of
time complexity In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...
.


Definitions

The natural logarithm can be defined in several equivalent ways.


Inverse of exponential

The most general definition is as the inverse function of e^x, so that e^=x. Because e^x is positive and invertible for any real input x, this definition of \ln(x) is well defined for any positive ''x''. For the
complex numbers 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 ...
, e^z is not invertible, so \ln(z) is a
multivalued 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 ...
. In order to make \ln(z) a proper, single-output
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 ...
, we therefore need to restrict it to a particular
principal branch In mathematics, a principal branch is a function which selects one branch ("slice") of a multi-valued function. Most often, this applies to functions defined on the complex plane. Examples Trigonometric inverses Principal branches are used ...
, often denoted by \operatorname(z). As the inverse function of e^z, \ln(z) can be defined by inverting the usual definition of e^z: :e^z = \lim_\left(1+\frac\right)^n Doing so yields: :\ln(z) = \lim_n\cdot (\sqrt 1) This definition therefore derives its own principal branch from the principal branch of nth roots.


Integral definition

The natural logarithm of a positive, real number may be defined as the area under the graph of the
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 ...
with equation between and . This is 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 ...
:\ln a = \int_1^a \frac\,dx. If is less than , then this area is considered to be negative. This function is a logarithm because it satisfies the fundamental multiplicative property of a logarithm: :\ln(ab) = \ln a + \ln b. This can be demonstrated by splitting the integral that defines into two parts, and then making the variable substitution (so ) in the second part, as follows: :\begin \ln ab = \int_1^\frac \, dx &=\int_1^a \frac \, dx + \int_a^ \frac \, dx\\ pt &=\int_1^a \frac 1 x \, dx + \int_1^b \frac a\,dt\\ pt &=\int_1^a \frac 1 x \, dx + \int_1^b \frac \, dt\\ pt &= \ln a + \ln b. \end In elementary terms, this is simply scaling by in the horizontal direction and by in the vertical direction. Area does not change under this transformation, but the region between and is reconfigured. Because the function is equal to the function , the resulting area is precisely . The number can then be defined to be the unique real number such that . The natural logarithm also has an improper integral representation, which can be derived with
Fubini's theorem In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if the ...
as follows: \ln\left(x\right)=\int_1^x \frac du = \int_1^x \int_0^\infty e^\ dt\ du = \int_0^\infty \int_1^x e^\ du\ dt = \int_^\fracdt


Properties

* \ln 1 = 0 * \ln e = 1 * \ln(xy) = \ln x + \ln y \quad \text\; x > 0\;\text\; y > 0 * \ln(x/y)= \ln x - \ln y * \ln(x^y) = y \ln x \quad \text\; x > 0 * \ln x < \ln y \quad\text\; 0 < x < y * \lim_ \frac = 1 * \lim_ \frac = \ln x\quad \text\; x > 0 * \frac \leq \ln x \leq x-1 \quad\text\quad x > 0 * \ln \leq \alpha x \quad\text\quad x \ge 0\;\text\; \alpha \ge 1 The statement is true for x=0, and we now show that \frac \ln \leq \frac ( \alpha x ) for all x, which completes the proof by 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 ...
. Hence, we want to show that :\frac \ln = \frac \leq \alpha = \frac ( \alpha x ) (Note that we have not yet proved that this statement is true.) If this is true, then by multiplying the middle statement by the positive quantity (1+x^\alpha) / \alpha and subtracting x^\alpha we would obtain : x^ \leq x^\alpha + 1 : x^ (1-x) \leq 1 This statement is trivially true for x \ge 1 since the left hand side is negative or zero. For 0 \le x < 1 it is still true since both factors on the left are less than 1 (recall that \alpha \ge 1). Thus this last statement is true and by repeating our steps in reverse order we find that \frac \ln \leq \frac ( \alpha x ) for all x. This completes the proof. An alternate proof is to observe that (1+x^\alpha)\leq (1+x)^\alpha under the given conditions. This can be proved, e.g., by the norm inequalities. Taking logarithms and using \ln(1+x)\leq x completes the proof.


Derivative

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 the natural logarithm as a real-valued function on the positive reals is given by :\frac \ln x = \frac. How to establish this derivative of the natural logarithm depends on how it is defined firsthand. If the natural logarithm is defined as the integral :\ln x = \int_1^x \frac\,dt, then the derivative immediately follows from the first part 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 ...
. On the other hand, if the natural logarithm is defined as the inverse of the (natural) exponential function, then the derivative (for ''x'' > 0) can be found by using the properties of the logarithm and a definition of the exponential function. From the definition of the number e = \lim_(1+u)^, the exponential function can be defined as e^x = \lim_(1+u)^ = \lim_(1 + hx)^ , where u=hx, h=u/x. The derivative can then be found from first principles. :\begin \frac \ln x &= \lim_ \frac \\ &= \lim_\left \frac \ln\left(\frac\right)\right\\ &= \lim_\left \ln\left(\left(1 + \frac\right)^\right )\rightquad &&\text\\ &= \ln \left \lim_\left(1 + \frac\right)^\rightquad &&\text \\ &= \ln e^ \quad &&\text e^x = \lim_(1 + hx)^\\ &= \frac \quad &&\text \end Also, we have: :\frac \ln ax = \frac (\ln a + \ln x) = \frac \ln a +\frac \ln x = \frac. so, unlike its inverse function e^, a constant in the function doesn't alter the differential.


Series

Since the natural logarithm is undefined at 0, \ln(x) itself does not have a
Maclaurin series Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin ( ...
, unlike many other elementary functions. Instead, one looks for Taylor expansions around other points. For example, if \vert x - 1 \vert \leq 1 \text x \neq 0, then :\begin \ln x &= \int_1^x \frac \, dt = \int_0^ \frac \, du \\ &= \int_0^ (1 - u + u^2 - u^3 + \cdots) \, du \\ &= (x - 1) - \frac + \frac - \frac + \cdots \\ &= \sum_^\infty \frac. \end This 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 ...
for ln ''x'' around 1. A change of variables yields the
Mercator series In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: :\ln(1+x)=x-\frac+\frac-\frac+\cdots In summation notation, :\ln(1+x)=\sum_^\infty \frac x^n. The series converges to the natural ...
: :\ln(1+x)=\sum_^\infty \frac x^k = x - \frac + \frac - \cdots, valid for , ''x'',  ≤ 1 and ''x'' ≠ −1.
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 ...
, disregarding x\ne -1, nevertheless applied this series to ''x'' = −1, in order to show that the harmonic series equals the (natural) logarithm of 1/(1 − 1), that is, the logarithm of infinity. Nowadays, more formally, one can prove that the harmonic series truncated at ''N'' is close to the logarithm of ''N'', when ''N'' is large, with the difference converging to 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 ...
. At right is a picture of ln(1 + ''x'') and some of its
Taylor polynomial 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 ...
s around 0. These approximations converge to the function only in the region −1 < ''x'' ≤ 1; outside of this region the higher-degree Taylor polynomials evolve to ''worse'' approximations for the function. A useful special case for positive integers ''n'', taking x=\tfrac, is: : \ln \left(\frac\right) = \sum_^\infty \frac = \frac - \frac + \frac - \frac + \cdots If \operatorname(x) \ge 1/2, then :\begin \ln (x) &= - \ln \left(\frac\right) = - \sum_^\infty \frac = \sum_^\infty \frac \\ &= \frac + \frac + \frac + \frac + \cdots \end Now, taking x=\tfrac for positive integers ''n'', we get: : \ln \left(\frac\right) = \sum_^\infty \frac = \frac + \frac + \frac + \frac + \cdots If \operatorname(x) \ge 0 \text x \neq 0, then : \ln (x) = \ln \left(\frac\right) = \ln\left(\frac\right) = \ln \left(1 + \frac\right) - \ln \left(1 - \frac\right). Since :\begin \ln(1+y) - \ln(1-y)&= \sum^\infty_\frac\left((-1)^y^i - (-1)^(-y)^i\right) = \sum^\infty_\frac\left((-1)^ +1\right) \\ &= y\sum^\infty_\frac\left((-1)^ +1\right)\overset\; 2y\sum^\infty_\frac, \end we arrive at :\begin \ln (x) &= \frac \sum_^\infty \frac ^k \\ &= \frac \left( \frac + \frac \frac + \frac ^2 + \cdots \right) . \end Using the substitution x=\tfrac again for positive integers ''n'', we get: :\begin \ln \left(\frac\right) &= \frac \sum_^\infty \frac\\ &= 2 \left(\frac + \frac + \frac + \cdots \right). \end This is, by far, the fastest converging of the series described here. The natural logarithm can also be expressed as an infinite product: :\ln(x)=(x-1) \prod_^\infty \left ( \frac \right ) Two examples might be: :\ln(2)=\left ( \frac \right )\left ( \frac \right )\left ( \frac \right )\left ( \frac \right )... :\pi=(2i+2)\left ( \frac \right )\left ( \frac \right )\left ( \frac \right )\left ( \frac \right )... From this identity, we can easily get that: :\frac=\frac-\sum_^\infty\frac For example: :\frac=2-\frac-\frac-\frac \cdots


The natural logarithm in integration

The natural logarithm allows simple
integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...
of functions of the form ''g''(''x'') = ''f'' '(''x'')/''f''(''x''): an
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 ''g''(''x'') is given by ln(, ''f''(''x''), ). This is the case because of 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 , ...
and the following fact: :\frac\ln \left, x \ = \frac. In other words, if x is a real number with x\not=0, then :\int \frac \,dx = \ln, x, + C and :\int = \ln, f(x), + C. Here is an example in the case of ''g''(''x'') = tan(''x''): : \begin & \int \tan x \,dx = \int \frac \,dx \\ pt& \int \tan x \,dx = \int \frac \,dx. \end Letting ''f''(''x'') = cos(''x''): :\int \tan x \,dx = -\ln \left, \cos x \ + C :\int \tan x \,dx = \ln \left, \sec x \ + C where ''C'' is an
arbitrary constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the Set (mathematics), set of all antiderivatives of f(x) ...
. The natural logarithm can be integrated using
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
: :\int \ln x \,dx = x \ln x - x + C. Let: :u = \ln x \Rightarrow du = \frac :dv = dx \Rightarrow v = x then: : \begin \int \ln x \,dx & = x \ln x - \int \frac \,dx \\ & = x \ln x - \int 1 \,dx \\ & = x \ln x - x + C \end


Efficient computation

For ln(''x'') where ''x'' > 1, the closer the value of ''x'' is to 1, the faster the rate of convergence of its Taylor series centered at 1. The identities associated with the logarithm can be leveraged to exploit this: :\begin \ln 123.456 &= \ln(1.23456 \cdot 10^2)\\ &= \ln 1.23456 + \ln(10^2)\\ &= \ln 1.23456 + 2 \ln 10\\ &\approx \ln 1.23456 + 2 \cdot 2.3025851. \end Such techniques were used before calculators, by referring to numerical tables and performing manipulations such as those above.


Natural logarithm of 10

The natural logarithm of 10, which has the decimal expansion 2.30258509..., plays a role for example in the computation of natural logarithms of numbers represented in
scientific notation Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, o ...
, as a mantissa multiplied by a power of 10: : \ln(a\cdot 10^n) = \ln a + n \ln 10. This means that one can effectively calculate the logarithms of numbers with very large or very small
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
using the logarithms of a relatively small set of decimals in the range .


High precision

To compute the natural logarithm with many digits of precision, the Taylor series approach is not efficient since the convergence is slow. Especially if is near 1, a good alternative is to use
Halley's method In numerical analysis, Halley's method is a root-finding algorithm used for functions of one real variable with a continuous second derivative. It is named after its inventor Edmond Halley. The algorithm is second in the class of Householder's met ...
or
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 ...
to invert the exponential function, because the series of the exponential function converges more quickly. For finding the value of to give using Halley's method, or equivalently to give using Newton's method, the iteration simplifies to : y_ = y_n + 2 \cdot \frac which has
cubic 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 ...
to . Another alternative for extremely high precision calculation is the formula :\ln x \approx \frac - m \ln 2, where denotes the arithmetic-geometric mean of 1 and , and :s = x 2^m > 2^, with chosen so that bits of precision is attained. (For most purposes, the value of 8 for m is sufficient.) In fact, if this method is used, Newton inversion of the natural logarithm may conversely be used to calculate the exponential function efficiently. (The constants ln 2 and π can be pre-computed to the desired precision using any of several known quickly converging series.) Or, the following formula can be used: :\ln x=\frac,\quad x\in (1,\infty) where :\theta_2(x)=\sum_x^, \quad\theta_3(x)=\sum_x^ are the
Jacobi theta functions In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...
. page 225 Based on a proposal by
William Kahan William "Velvel" Morton Kahan (born June 5, 1933) is a Canadian mathematician and computer scientist, who received the Turing Award in 1989 for "''his fundamental contributions to numerical analysis''", was named an ACM Fellow in 1994, and inducte ...
and first implemented in the
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HP-41C The HP-41C series are programmable, expandable, continuous memory handheld RPN calculators made by Hewlett-Packard from 1979 to 1990. The original model, HP-41C, was the first of its kind to offer alphanumeric display capabilities. Later cam ...
calculator in 1979 (referred to under "LN1" in the display, only), some calculators,
operating system An operating system (OS) is system software that manages computer hardware, software resources, and provides common services for computer programs. Time-sharing operating systems schedule tasks for efficient use of the system and may also in ...
s (for example
Berkeley UNIX 4.3BSD The Berkeley Software Distribution or Berkeley Standard Distribution (BSD) is a discontinued operating system based on Research Unix, developed and distributed by the Computer Systems Research Group (CSRG) at the University of California, Berke ...
),
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The de ...
s and programming languages (for example
C99 C99 (previously known as C9X) is an informal name for ISO/IEC 9899:1999, a past version of the C programming language standard. It extends the previous version ( C90) with new features for the language and the standard library, and helps impl ...
) provide a special natural logarithm plus 1 function, alternatively named LNP1,Searchable PDF
/ref> or log1p to give more accurate results for logarithms close to zero by passing arguments ''x'', also close to zero, to a function log1p(''x''), which returns the value ln(1+''x''), instead of passing a value ''y'' close to 1 to a function returning ln(''y''). The function log1p avoids in the floating point arithmetic a near cancelling of the absolute term 1 with the second term from the Taylor expansion of the ln. This keeps the argument, the result, and intermediate steps all close to zero where they can be most accurately represented as floating-point numbers. In addition to base the
IEEE 754-2008 The Institute of Electrical and Electronics Engineers (IEEE) is a 501(c)(3) professional association for electronic engineering and electrical engineering (and associated disciplines) with its corporate office in New York City and its operation ...
standard defines similar logarithmic functions near 1 for
binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that t ...
and
decimal 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: and . Similar inverse functions named "
expm1 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, al ...
", "expm" or "exp1m" exist as well, all with the meaning of .For a similar approach to reduce
round-off error A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are d ...
s of calculations for certain input values see
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 like
versine The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit Āryabhaṭa's sine table , ''Aryabhatia'',
,
vercosine The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'',coversine The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'',covercosine The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'',haversine The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'',havercosine The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit Āryabhaṭa's sine table , ''Aryabhatia'',
,
hacoversine The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'',hacovercosine The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'',exsecant The exsecant (exsec, exs) and excosecant (excosec, excsc, exc) are trigonometric functions defined in terms of the secant and cosecant functions. They used to be important in fields such as surveying, railway engineering, civil engineering, astro ...
and
excosecant The exsecant (exsec, exs) and excosecant (excosec, excsc, exc) are trigonometric functions defined in terms of the secant and cosecant functions. They used to be important in fields such as surveying, railway engineering, civil engineering, astr ...
.
An identity in terms of 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 s ...
, :\mathrm(x) = \log(1+x) = 2 ~ \mathrm\left(\frac\right)\,, gives a high precision value for small values of on systems that do not implement .


Computational complexity

The
computational complexity In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) ...
of computing the natural logarithm using the arithmetic-geometric mean (for both of the above methods) is O(''M''(''n'') ln ''n''). Here ''n'' is the number of digits of precision at which the natural logarithm is to be evaluated and ''M''(''n'') is the computational complexity of multiplying two ''n''-digit numbers.


Continued fractions

While no simple
continued fraction In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...
s are available, several
generalized continued fraction In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values. A gen ...
s are, including: : \begin \ln(1+x) & =\frac-\frac+\frac-\frac+\frac-\cdots \\ pt& = \cfrac \end : \begin \ln\left(1+\frac\right) & = \cfrac \\ pt& = \cfrac \end These continued fractions—particularly the last—converge rapidly for values close to 1. However, the natural logarithms of much larger numbers can easily be computed, by repeatedly adding those of smaller numbers, with similarly rapid convergence. For example, since 2 = 1.253 × 1.024, the
natural logarithm of 2 The decimal value of the natural logarithm of 2 is approximately :\ln 2 \approx 0.693\,147\,180\,559\,945\,309\,417\,232\,121\,458. The logarithm of 2 in other bases is obtained with the formula :\log_b 2 = \frac. The common logarithm in particu ...
can be computed as: : \begin \ln 2 & = 3 \ln\left(1+\frac\right) + \ln\left(1+\frac\right) \\ pt& = \cfrac + \cfrac . \end Furthermore, since 10 = 1.2510 × 1.0243, even the natural logarithm of 10 can be computed similarly as: : \begin \ln 10 & = 10 \ln\left(1+\frac\right) + 3\ln\left(1+\frac\right) \\
0pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
& = \cfrac + \cfrac . \end The reciprocal of the natural logarithm can be also written in this way: :\frac =\frac \sqrt\sqrt\ldots For example: :\frac =\frac \sqrt\sqrt\ldots


Complex logarithms

The exponential function can be extended to a function which gives a
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 ...
as for any arbitrary complex number ; simply use the infinite series with =z complex. This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm. There are two difficulties involved: no has ; and it turns out that . Since the multiplicative property still works for the complex exponential function, , for all complex and integers . So the logarithm cannot be defined for the whole
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 ...
, and even then it is
multi-valued 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 ...
—any complex logarithm can be changed into an "equivalent" logarithm by adding any integer multiple of at will. The complex logarithm can only be single-valued on the cut plane. For example, or or , etc.; and although can be defined as , or or , and so on. principal branch In mathematics, a principal branch is a function which selects one branch ("slice") of a multi-valued function. Most often, this applies to functions defined on the complex plane. Examples Trigonometric inverses Principal branches are used ...
)"> Image:NaturalLogarithmRe.png, Image:NaturalLogarithmImAbs.png, Image:NaturalLogarithmAbs.png, Image:NaturalLogarithmAll.png, Superposition of the previous three graphs


See also

* Approximating natural exponents (log base e) *
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 ...
*
Napierian logarithm The term Napierian logarithm or Naperian logarithm, named after John Napier, is often used to mean the natural logarithm. Napier did not introduce this ''natural'' logarithmic function, although it is named after him. However, if it is taken to m ...
* List of logarithmic identities *
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 ...
*
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 (\ ...
*
Logarithmic integral function 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 a ...
*
Nicholas Mercator Nicholas (Nikolaus) Mercator (c. 1620, Holstein – 1687, Versailles), also known by his German name Kauffmann, was a 17th-century mathematician. He was born in Eutin, Schleswig-Holstein, Germany and educated at Rostock and Leyden after which he ...
– first to use the term natural logarithm *
Polylogarithm In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natur ...
*
Von Mangoldt function In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive. Definition The von Mangold ...


Notes


References

{{Calculus topics Logarithms Elementary special functions E (mathematical constant) Unary operations de:Logarithmus#Natürlicher Logarithmus