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*logarithmic scale*

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logarithmic scale

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**logarithmic scale**is a nonlinear scale used when there is a large range of quantities. Common uses include earthquake strength, sound loudness, light intensity, and pH of solutions.It is based on orders of magnitude, rather than a standard linear scale, so the value represented by each equidistant mark on the scale is the value at the previous mark multiplied by a constant.Logarithmic scales are also used in slide rules for multiplying or dividing numbers by adding or subtracting lengths on the scales.(File:slide rule example3.svg|frame|center|The two logarithmic scales of a slide rule)

## Common usages

(File:Internet host count 1988-2012 log scale.png|thumb|Graph on a logarithmic scale)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 and movement in the earth
- sound level, with units bel and decibel
- neper 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
- logit for odds in statistics
- Palermo Technical Impact Hazard Scale
- logarithmic timeline
- counting f-stops for ratios of photographic exposure
- the rule of 'nines' used for rating low probabilities
- entropy in thermodynamics
- information in information theory
- particle-size-distribution curves of soil

- pH for acidity
- stellar magnitude scale for brightness of stars
- Krumbein scale for particle size in geology
- absorbance of light by transparent samples

## Graphic representation

File:Logarithmic Scales.svg|thumb|400px|Various scales: linâ€“lin, linâ€“log, logâ€“lin, and (Logâ€“log plot|logâ€“log]]. Plotted graphs are:*y*= 10

*x*(red),

*y*=

*x*(green),

*y*= log

*e*(

*x*) (blue).)The top left graph is linear in the X and Y axis, 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.

### Logâ€“log plots

(File:Log-log plot example.svg|thumb|200px|Plot on logâ€“log scale of equation of a line.)If both the vertical and horizontal axes of a plot are scaled logarithmically, the plot is referred to as a logâ€“log plot.### Semi-logarithmic plots

If only the ordinate or abscissa is scaled logarithmically, the plot is referred to as a semi-logarithmic plot.## Logarithmic units

A**logarithmic unit**is an abstract mathematical unit that can be used to express any quantity (physical or mathematical) that is defined on a logarithmic scale, that is, as being proportional to the value of a logarithm function. Here, a given logarithmic unit will be denoted using the notation [log

*n*], where

*n*is a positive real number, and [log ] here denotes the indefinite logarithm function Log().

### Examples

Examples of logarithmic units include common units of information and entropy, such as the*bit*[log 2]{{Dubious|date=March 2016}} and the

*byte*8[log 2] = [log 256], also the

*nat*[log e] and the

*ban*[log 10]; units of relative signal strength magnitude such as the

*decibel*0.1[log 10] and

*bel*[log 10],

*neper*[log e], and other logarithmic-scale units such as the Richter magnitude scale point [log 10] or (more generally) the corresponding order-of-magnitude unit sometimes referred to as a

*factor of ten*or

*decade*(here meaning [log 10], not 10 years). Musical pitch intervals are also logarithmic units on a frequency scale, such as octave [log 2], semitone, cent, etc.

### Motivation

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
log_b a = frac{log_c a}{log_c b},

the logarithms of any given number *a*to two different bases (here

*b*and

*c*) differ only by the constant factor log

*c*

*b*. This constant factor can be considered to represent the conversion factor for converting a numerical representation of the pure (indefinite) logarithmic quantity Log(

*a*) from one arbitrary unit of measurement (the [log

*c*] unit) to another (the [log

*b*] unit), since

operatorname{Log}(a) = (log_b a)[log b] = (log_c a)[log c] .

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

*S*= Log(

*W*), where "Log" here denotes the indefinite logarithm, and we let

*k*= [log e]; that is, we identify the physical entropy unit

*k*with the mathematical unit [log e]. This identity works because

ln W = log_e W = frac{operatorname{Log}(W)}{log e}.

Thus, we can interpret Boltzmann's constant as being simply the expression (in terms of more standard physical units) of the abstract logarithmic unit [log 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 Log(

*W*), which implies no particular choice of base, and thus no particular choice of physical unit for measuring entropy.Algebraically, a choice of logarithmic unit also allows one to define a "logarithmic addition" on the log scale, so together with "logarithmic multiplication" (usual addition) it has the structure of a semiring, known as the log semiring. This has the interesting property that as the scale grows to infinity, logarithmic addition converges to maximum. The study of this phenomenon is known as tropical analysis.

## See also

- Bode plot
- John Napier
- Level (logarithmic quantity)
- Logarithm
- Logarithmic mean
- Log semiring
- Multiplicative calculus
- Preferred number

### Units of information

### Units of relative amplitude or power

### Scale

### Applications

## References

{{Reflist}}## Further reading

- JOURNAL, Stanislas, Dehaene, VÃ©ronique, Izard, Elizabeth, Spelke, Elizabeth Spelke, Pierre, Pica, Pierre Pica, 2008, Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures, Science, 320, 5880, 10.1126/science.1156540, 18511690, 2610411, 1217â€“20, 2008Sci...320.1217D,
- JOURNAL, Karl, Tuffentsammer, P., Schumacher, Normzahlen â€“ die einstellige Logarithmentafel des Ingenieurs, German, Preferred numbers - the engineer's single-digit logarithm table, Werkstattechnik und Maschinenbau, 43, 4, 1953, 156,
- JOURNAL, Tuffentsammer, Karl, Das Dezilog, eine BrÃ¼cke zwischen Logarithmen, Dezibel, Neper und Normzahlen, German, The decilog, a bridge between logarithms, decibel, neper and preferred numbers, VDI-Zeitschrift, 98, 1956, 267â€“274,
- BOOK, Clemens, Ries, Normung nach Normzahlen, German, Standardization by preferred numbers, {{ill, Duncker & Humblot Verlag, de, |location=Berlin, Germany |date=1962 |edition=1 |isbn=978-3-42801242-8}} (135 pages)
- BOOK, Logarithmen, Normzahlen, Dezibel, Neper, Phon - natÃ¼rlich verwandt!, German, Logarithms, preferred numbers, decibel, neper, phon - naturally related!, Eugen, Paulin, 2007-09-01,weblink 2016-12-18, no,weblink" title="web.archive.org/web/20161218223050weblink">weblink 2016-12-18,

## External links

- WEB,weblink GNU Emacs Calc Manual: Logarithmic Units, Gnu.org, 2016-11-23,
- Non-Newtonian calculus website

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