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
: 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.
The markings on slide rule
s 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
in the Earth
* Sound level
, with units decibel
for amplitude, field and power quantities
* Frequency level
, with units cent
, minor second
, major second
, and octave
for the relative pitch of notes in music
* Palermo Technical Impact Hazard Scale
* Logarithmic timeline
* Counting f-stop
s for ratios of photographic exposure
* The rule of 'nines'
used for rating low probabilities
in information theory
* 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:
* Stellar magnitude scale
for brightness of star
* Krumbein scale
for particle size
of light by transparent samples
Some of our sense
s 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.
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
s or power law
s, since these will show up as straight lines.
A slide rule
has logarithmic scales, and nomogram
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.
If both the vertical and horizontal axes of a plot are scaled logarithmically, the plot is referred to as a log–log plot
If only the ordinate
is scaled logarithmically, the plot is referred to as a semi-logarithmic plot.
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 of logarithmic units include units of data storage capacity
), of information
and information entropy
), and of signal level
, bel, neper
). Logarithmic frequency quantities are used in electronics (decade
) and for music pitch interval
, etc.). Other logarithmic scale units include the Richter magnitude scale
In addition, several industrial measures are logarithmic, such as standard values for resistors
, the American wire gauge
, the Birmingham gauge
used for wire and needles, and so on.
Units of information
Units of level or level difference
Units of frequency interval
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
The motivation behind the concept of logarithmic units is that defining a quantity on a logarithmic scale in terms of a logarithm to a specific base amounts to making a (totally arbitrary) choice of a unit of measurement for that quantity, one that corresponds to the specific (and equally arbitrary) logarithm base that was selected. Due to the identity
the logarithms of any given number ''a'' to two different bases (here ''b'' and ''c'') differ only by the constant factor
. This constant factor can be considered to represent the conversion factor for converting a numerical representation of the pure (indefinite) logarithmic quantity
from one arbitrary unit of measurement (the og ''c''
unit) to another (the og ''b''
For example, Boltzmann
's standard definition of entropy ''S'' = ''k'' ln ''W'' (where ''W'' is the number of ways of arranging a system and ''k'' is Boltzmann's constant
) can also be written more simply as just
. where "Log" denotes the indefinite logarithm, and ''k'' = og e
that is, the physical entropy unit ''k'' can be identified with the mathematical unit og e
This identity works because
Thus, Boltzmann's constant can be interprested as being simply the expression (in terms of more standard physical units) of the abstract logarithmic unit og e
that is needed to convert the dimensionless pure-number quantity ln ''W'' (which uses an arbitrary choice of base, namely e) to the more fundamental pure logarithmic quantity
, which implies no particular choice of base, and thus no particular choice of physical unit for measuring entropy.
* Alexander Graham Bell
* Bode plot
* John Napier
* Level (logarithmic quantity)
* Logarithmic mean
* Log semiring
* Preferred number
* Order of magnitude
* Entropy (information theory)
* Richter magnitude scale
* (135 pages)
Non-Newtonian calculus website