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
Scale or scales may refer to:
Mathematics
* Scale (descriptive set theory), an object defined on a set of points
* Scale (ratio), the ratio of a linear dimension of a model to the corresponding dimension of the original
* Scale factor, a number ...
is
nonlinear: 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 curves are displayed on a log scale, otherwise they would increase too quickly to fit within a small
graph. 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 rules are arranged in a log scale for multiplying or dividing numbers by adding or subtracting lengths on the scales.

The following are examples of commonly used logarithmic scales, where a larger quantity results in a higher value:
*
Richter magnitude scale and
moment magnitude scale (MMS) for strength of
earthquakes
An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in intensity, from ...
and
movement
Movement may refer to:
Common uses
* Movement (clockwork), the internal mechanism of a timepiece
* Motion, commonly referred to as movement
Arts, entertainment, and media
Literature
* "Movement" (short story), a short story by Nancy Fu ...
in the
Earth 
*
Sound level, with units
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 ...
*
Neper
The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. As ...
for amplitude, field and power quantities
*
Frequency level
In science and engineering, a power level and a field level (also called a root-power level) are logarithmic measures of certain quantities referenced to a standard reference value of the same type.
* A ''power level'' is a logarithmic quantity ...
, with units
cent
Cent may refer to:
Currency
* Cent (currency), a one-hundredth subdivision of several units of currency
* Penny (Canadian coin), a Canadian coin removed from circulation in 2013
* 1 cent (Dutch coin), a Dutch coin minted between 1941 and 1944
* ...
,
minor second,
major second, and
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 ...
for the relative pitch of notes in
music
*
Logit for
odds in
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
*
Palermo Technical Impact Hazard Scale
*
Logarithmic timeline
* Counting
f-stops for ratios of
photographic exposure
* The rule of nines used for rating low
probabilities
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
*
Entropy in
thermodynamics
*
Information 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 ...
* Particle size distribution curves of soil

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
star
A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
s
*
Krumbein scale for
particle size in
geology
*
Absorbance of light by transparent samples
Some of our
senses operate in a logarithmic fashion (
Weber–Fechner law), which makes logarithmic scales for these input quantities especially appropriate. In particular, our sense of
hearing 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

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 law
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 q ...
s or
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, since these will show up as straight lines.
A
slide rule has logarithmic scales, and
nomograms often employ logarithmic scales. The
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 two numbers is midway between the numbers. Before the advent of computer graphics, logarithmic
graph paper
Graph paper, coordinate paper, grid paper, or squared paper is writing paper that is printed with fine lines making up a regular grid. The lines are often used as guides for plotting graphs of functions or experimental data and drawing curves. I ...
was a commonly used scientific tool.
Log–log plots

If both the vertical and horizontal axes of a plot are scaled logarithmically, the plot is referred to as a
log–log plot.
Semi-logarithmic plots
If only the
ordinate or
abscissa is scaled logarithmically, the plot is referred to as a
semi-logarithmic 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 that can be used to express a quantity (
physical or mathematical) on a logarithmic scale, that is, as being proportional to the value of a
logarithm 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
Computer data storage is a technology consisting of computer components and recording media that are used to retain digital data. It is a core function and fundamental component of computers.
The central processing unit (CPU) of a compu ...
(
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 represented a ...
,
byte), of
information and
information entropy (
nat,
shannon,
ban
Ban, or BAN, may refer to:
Law
* Ban (law), a decree that prohibits something, sometimes a form of censorship, being denied from entering or using the place/item
** Imperial ban (''Reichsacht''), a form of outlawry in the medieval Holy Roman ...
), and of
signal level (
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 ...
, bel,
neper
The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. As ...
). Logarithmic frequency quantities are used in electronics (
decade,
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 ...
) and for music pitch
intervals (
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 ...
,
semitone,
cent
Cent may refer to:
Currency
* Cent (currency), a one-hundredth subdivision of several units of currency
* Penny (Canadian coin), a Canadian coin removed from circulation in 2013
* 1 cent (Dutch coin), a Dutch coin minted between 1941 and 1944
* ...
, etc.). Other logarithmic scale units include the
Richter magnitude scale point.
In addition, several industrial measures are logarithmic, such as standard values for
resistors
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
, the
American wire gauge, the
Birmingham gauge 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 represented a ...
,
byte
*
hartley
Hartley may refer to:
Places Australia
*Hartley, New South Wales
*Hartley, South Australia
**Electoral district of Hartley, a state electoral district
Canada
*Hartley Bay, British Columbia
United Kingdom
*Hartley, Cumbria
*Hartley, Plymou ...
*
nat
*
shannon
Units of level or level difference
*
bel BEL can be an abbreviation for:
* The ISO 3166-1 alpha-3 country code for Belgium
* ''BEL'' or bell character in the C0 control code set
* Belarusian language, in the ISO 639-2 and SIL country code lists
* Bharat Electronics Limited, an Indian stat ...
,
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 ...
*
neper
The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. As ...
Units of frequency interval
*
decade,
decidecade,
savart
*
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 ...
,
tone,
semitone,
cent
Cent may refer to:
Currency
* Cent (currency), a one-hundredth subdivision of several units of currency
* Penny (Canadian coin), a Canadian coin removed from circulation in 2013
* 1 cent (Dutch coin), a Dutch coin minted between 1941 and 1944
* ...
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
Alexander Graham Bell (, born Alexander Bell; March 3, 1847 – August 2, 1922) was a Scottish-born inventor, scientist and engineer who is credited with patenting the first practical telephone. He also co-founded the American Telephone and Te ...
*
Bode plot
*
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 ...
(arithmetic mean in logscale)
*
John Napier
*
Level (logarithmic quantity)
In science and engineering, a power level and a field level (also called a root-power level) are logarithmic measures of certain quantities referenced to a standard reference value of the same type.
* A ''power level'' is a logarithmic quantity ...
*
Logarithm
*
Logarithmic mean
*
Log semiring
*
Preferred number
*
Semi-log plot
Scale
*
Order of magnitude
Applications
*
Entropy
*
Entropy (information theory)
*
pH
*
Richter magnitude scale
References
Further reading
*
*
*
* (135 pages)
*
External links
*
Non-Newtonian calculus website
{{DEFAULTSORT:Logarithmic Scale
Non-Newtonian calculus