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
scale is
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
: the numbers 10 and 20, and 60 and 70, are not the same distance apart on a log scale. Rather, the numbers 10 and 100, and 60 and 600 are equally spaced. Thus moving a unit of distance along the scale means the number has been ''multiplied'' by 10 (or some other fixed factor). Often
exponential growth
Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a ...
curves are displayed on a log scale, otherwise they would increase too quickly to fit within a small
graph
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 ...
. Another way to think about it is that the ''number of
digits'' of the data grows at a constant rate. For example, the numbers 10, 100, 1000, and 10000 are equally spaced on a log scale, because their numbers of digits is going up by 1 each time: 2, 3, 4, and 5 digits. In this way, adding two digits ''multiplies'' the quantity measured on the log scale by a factor of 100.
Common uses
The markings on
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 ...
s are arranged in a log scale for multiplying or dividing numbers by adding or subtracting lengths on the scales.
![slide rule example3](https://upload.wikimedia.org/wikipedia/commons/1/15/Slide_rule_example3.svg)
The following are examples of commonly used logarithmic scales, where a larger quantity results in a higher value:
*
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 ...
and
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 ...
(MMS) for strength of
earthquakes and
movement in the
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
![COB data Tsunami deaths](https://upload.wikimedia.org/wikipedia/commons/e/e0/COB_data_Tsunami_deaths.PNG)
*
Sound level, with units
decibel
*
Neper for amplitude, field and power quantities
*
Frequency level, with units
cent,
minor second
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 ...
,
major second
In Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a musical interval encompassing two adjacent staff positions (see Interval number for more de ...
, and
octave for the relative pitch of notes in
music
Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspe ...
*
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 ...
for
odds
Odds provide a measure of the likelihood of a particular outcome. They are calculated as the ratio of the number of events that produce that outcome to the number that do not. Odds are commonly used in gambling and statistics.
Odds also have ...
in
statistics
*
Palermo Technical Impact Hazard Scale
The Palermo Technical Impact Hazard Scale is a logarithmic scale used by astronomers to rate the potential hazard of impact of a near-Earth object (NEO). It combines two types of data—probability of impact and estimated kinetic yield— ...
*
Logarithmic timeline
A logarithmic timeline is a timeline laid out according to a logarithmic scale. This necessarily implies a zero point and an infinity point, neither of which can be displayed. The most natural zero point is the Big Bang, looking forward, but ...
* Counting
f-stop
In optics, the f-number of an optical system such as a camera lens is the ratio of the system's focal length to the diameter of the entrance pupil ("clear aperture").Smith, Warren ''Modern Optical Engineering'', 4th Ed., 2007 McGraw-Hill ...
s for ratios of
photographic exposure
In photography, exposure is the amount of light per unit area (the image plane's illuminance times the exposure time) reaching a frame of photographic film or the surface of an electronic image sensor, as determined by shutter speed, lens ...
* The rule of nines used for rating low
probabilities
*
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 ...
in
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of th ...
*
Information
Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random ...
in
information theory
* Particle size distribution curves of soil
![Solarmap](https://upload.wikimedia.org/wikipedia/commons/2/29/Solarmap.gif)
The following are examples of commonly used logarithmic scales, where a larger quantity results in a lower (or negative) value:
*
pH for acidity
*
Stellar magnitude scale for brightness of
stars
*
Krumbein scale
Grain size (or particle size) is the diameter of individual grains of sediment, or the lithification, lithified particles in clastic rocks. The term may also be applied to other granular materials. This is different from the crystallite si ...
for
particle size
Particle size is a notion introduced for comparing dimensions of solid particles ('' flecks''), liquid particles ('' droplets''), or gaseous particles ('' bubbles''). The notion of particle size applies to particles in colloids, in ecology, in ...
in
geology
Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other Ea ...
*
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 by transparent samples
Some of our
sense
A sense is a biological system used by an organism for sensation, the process of gathering information about the world through the detection of stimuli. (For example, in the human body, the brain which is part of the central nervous system re ...
s operate in a logarithmic fashion (
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 an ...
), which makes logarithmic scales for these input quantities especially appropriate. In particular, our sense of
hearing
Hearing, or auditory perception, is the ability to perceive sounds through an organ, such as an ear, by detecting vibrations as periodic changes in the pressure of a surrounding medium. The academic field concerned with hearing is audit ...
perceives equal ratios of frequencies as equal differences in pitch. In addition, studies of young children in an isolated tribe have shown logarithmic scales to be the most natural display of numbers in some cultures.
Graphic representation
![Logarithmic Scales-mkII](https://upload.wikimedia.org/wikipedia/commons/7/7e/Logarithmic_Scales-mkII.svg)
The top left graph is linear in the X and Y axes, and the Y-axis ranges from 0 to 10. A base-10 log scale is used for the Y axis of the bottom left graph, and the Y axis ranges from 0.1 to 1,000.
The top right graph uses a log-10 scale for just the X axis, and the bottom right graph uses a log-10 scale for both the X axis and the Y axis.
Presentation of data on a logarithmic scale can be helpful when the data:
* covers a large range of values, since the use of the logarithms of the values rather than the actual values reduces a wide range to a more manageable size;
* may contain
exponential laws or
power laws, since these will show up as straight lines.
A
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 ...
has logarithmic scales, and
nomogram
A nomogram (from Greek , "law" and , "line"), also called a nomograph, alignment chart, or abac, is a graphical calculating device, a two-dimensional diagram designed to allow the approximate graphical computation of a mathematical function ...
s often employ logarithmic scales. The
geometric mean of two numbers is midway between the numbers. Before the advent of computer graphics, logarithmic
graph paper was a commonly used scientific tool.
Log–log plots
![Log-log plot example](https://upload.wikimedia.org/wikipedia/commons/1/18/Log-log_plot_example.svg)
If both the vertical and horizontal axes of a plot are scaled logarithmically, the plot is referred to as a
log–log plot
In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Power functions – relationships of the form y=ax^k – appear a ...
.
Semi-logarithmic plots
If only the
ordinate
In common usage, the abscissa refers to the (''x'') coordinate and the ordinate refers to the (''y'') coordinate of a standard two-dimensional graph.
The distance of a point from the y-axis, scaled with the x-axis, is called abscissa or x c ...
or
abscissa
In common usage, the abscissa refers to the (''x'') coordinate and the ordinate refers to the (''y'') coordinate of a standard two-dimensional graph.
The distance of a point from the y-axis, scaled with the x-axis, is called abscissa or x coo ...
is scaled logarithmically, the plot is referred to as a
semi-logarithmic
In science and engineering, a semi-log plot/graph or semi-logarithmic plot/graph has one axis on a logarithmic scale, the other on a linear scale. It is useful for data with exponential relationships, where one variable covers a large range of v ...
plot.
Extensions
A modified log transform can be defined for negative input (''y''<0) and to avoid the singularity for zero input (''y''=0) so as to produce symmetric log plots:
:
for a constant ''C''=1/ln(10).
Logarithmic units
A logarithmic unit 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'' (a ...
that can be used to express a quantity (
physical
Physical may refer to:
*Physical examination
In a physical examination, medical examination, or clinical examination, a medical practitioner examines a patient for any possible medical signs or symptoms of a medical condition. It generally co ...
or mathematical) on a logarithmic scale, that is, as being proportional to the value of a
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 of ...
function applied to the ratio of the quantity and a reference quantity of the same type. The choice of unit generally indicates the type of quantity and the base of the logarithm.
Examples
Examples of logarithmic units include units of
data storage capacity (
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 ...
,
byte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable uni ...
), of
information
Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random ...
and
information entropy
In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable X, which takes values in the alphabet \ ...
(
nat,
shannon,
ban), and of
signal level
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 ...
(
decibel, bel,
neper). Logarithmic frequency quantities are used in electronics (
decade
A decade () is a period of ten years. Decades may describe any ten-year period, such as those of a person's life, or refer to specific groupings of calendar years.
Usage
Any period of ten years is a "decade". For example, the statement that "du ...
,
octave) and for music pitch
intervals (
octave,
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 ...
,
cent, etc.). Other logarithmic scale units include 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 ...
point.
In addition, several industrial measures are logarithmic, such as standard values for
resistors, the
American wire gauge, the
Birmingham gauge
The Birmingham gauge is a wire gauge system, and is also used to specify thickness or diameter of hypodermic needles and tube products.
Terminology
Birmingham gauge is also known as the Stubs Iron Wire Gauge or Birmingham Wire Gauge. It is not t ...
used for wire and needles, and so on.
Units of information
*
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 ...
,
byte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable uni ...
*
hartley
*
nat
*
shannon
Units of level or level difference
*
bel,
decibel
*
neper
Units of frequency interval
*
decade
A decade () is a period of ten years. Decades may describe any ten-year period, such as those of a person's life, or refer to specific groupings of calendar years.
Usage
Any period of ten years is a "decade". For example, the statement that "du ...
,
decidecade
A one-third octave is a logarithmic unit of frequency ratio equal to either one third of an octave (1200/3 = 400 cents: major third) or one tenth of a decade (3986.31/10 = 398.631 cents: M3 ). An alternative (unambiguous) term for one tenth of a ...
,
savart
The savart is a unit of measurement for musical pitch intervals (). One savart is equal to one thousandth of a decade ( 10/1: 3,986.313714 cents): 3.9863 cents. Musically, in just intonation, the interval of a decade is precisely a just major ...
*
octave,
tone,
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 ...
,
cent
Table of examples
The two definitions of a decibel are equivalent, because a ratio of
power quantities is equal to the square of the corresponding ratio of
root-power quantities.
See also
*
Alexander Graham Bell
*
Bode plot
*
Geometric mean (arithmetic mean in logscale)
*
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 Ioan ...
*
Level (logarithmic quantity)
*
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 of ...
*
Logarithmic mean
In mathematics, the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient.
This calculation is applicable in engineering problems involving heat and mass tran ...
*
Log semiring
In mathematics, in the field of tropical analysis, the log semiring is the semiring structure on the logarithmic scale, obtained by considering the extended real numbers as logarithms. That is, the operations of addition and multiplication are d ...
*
Preferred number
In industrial design, preferred numbers (also called preferred values or preferred series) are standard guidelines for choosing exact product dimensions within a given set of constraints.
Product developers must choose numerous lengths, distanc ...
*
Semi-log plot
In science and engineering, a semi-log plot/graph or semi-logarithmic plot/graph has one axis on a logarithmic scale, the other on a linear scale. It is useful for data with exponential relationships, where one variable covers a large range o ...
Scale
*
Order of magnitude
An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic di ...
Applications
*
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 ...
*
Entropy (information theory)
In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable X, which takes values in the alphabet \ ...
*
pH
*
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 ...
References
Further reading
*
*
*
* (135 pages)
*
External links
*
Non-Newtonian calculus website
{{DEFAULTSORT:Logarithmic Scale
Non-Newtonian calculus