Editing Logarithmic scale

{{Short description|Measurement scale based on orders of magnitude}}
[[File:Internet host count 1988-2012 log scale.png|thumb|upright=1.7|right|[[Semi-log plot]] of the Internet host count over time shown on a logarithmic scale]]
A '''logarithmic scale''' (or '''log scale''') is a method used to display numerical data that spans a broad range of values, especially when there are significant differences between the magnitudes of the numbers involved. 

Unlike a linear [[Scale (measurement)|scale]] where each unit of distance corresponds to the same increment, on a logarithmic scale each unit of length is a multiple of some base value raised to a power, and corresponds to the multiplication of the previous value in the scale by the base value. In common use, logarithmic scales are in base 10 (unless otherwise specified). 

A logarithmic scale is [[Nonlinear system|nonlinear]], and as such numbers with equal distance between them such as 1, 2, 3, 4, 5 are not equally spaced. Equally spaced values on a logarithmic scale have exponents that increment uniformly. Examples of equally spaced values are 10, 100, 1000, 10000, and 100000 (i.e., 10<sup>1</sup>, 10<sup>2</sup>, 10<sup>3</sup>, 10<sup>4</sup>, 10<sup>5</sup>) and 2, 4, 8, 16, and 32 (i.e., 2<sup>1</sup>, 2<sup>2</sup>, 2<sup>3</sup>, 2<sup>4</sup>, 2<sup>5</sup>). 

[[Exponential growth]] curves are often depicted on a logarithmic scale [[Plot (graphics)|graph]]. 
[[File:Logarithmic_scale.svg|thumb|upright=1.7|right|A logarithmic scale from 0.1 to 100]]
[[File:slide rule example3.svg|thumb|upright=1.7|right|The two logarithmic scales of a [[slide rule]]]]

== Common uses ==
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]] and [[Motion (physics)|movement]] in the [[Earth]]
[[File:COB data Tsunami deaths.PNG|right|thumb|upright=1.7|A logarithmic scale makes it easy to compare values that cover a large range, such as in this map.]]
* [[Sound level (disambiguation)|Sound level]], with the unit [[decibel]]
* [[Neper]] for amplitude, field and power quantities
* [[Frequency level]], with units [[Cent (music)|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-stop]]s 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

[[File:Solarmap.gif|thumb|upright=1.7|right|Map of the [[Solar System]] and the distance to [[Proxima Centauri]], using a logarithmic scale and measured in [[astronomical units]].]]
The following are examples of commonly used logarithmic scales, where a larger quantity results in a lower (or negative) value:
* [[pH]] for acidity
* [[Apparent magnitude|Stellar magnitude scale]] for brightness of [[star]]s
* [[Krumbein scale]] for [[Particle size (grain size)|particle size]] in [[geology]]
* [[Absorbance]] 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 (sense)|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.<ref>{{cite web|url=https://www.sciencedaily.com/releases/2008/05/080529141344.htm|title=Slide Rule Sense: Amazonian Indigenous Culture Demonstrates Universal Mapping Of Number Onto Space|date=2008-05-30|website=ScienceDaily|access-date=2008-05-31}}</ref> 
{{Clear}}

== Graphic representation ==
<!-- This section is linked from [[Order of magnitude]] -->
[[File:Logarithmic Scales-mkII.svg|thumb|upright=1.7|Various scales: lin–lin, [[Lin–log graph|lin–log, log–lin]], and [[Log–log plot|log–log]]. Plotted graphs are: ''y''&nbsp;=&nbsp;10<sup>&nbsp;''x''</sup> (<span style="color:red;">red</span>), ''y''&nbsp;=&nbsp;''x'' (<span style="color:green;">green</span>), ''y''&nbsp;=&nbsp;log<sub>''e''</sub>(''x'') (<span style="color:blue;">blue</span>).]]

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 1000.

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.

=== Log–log plots ===
{{Main|Log–log plot}}
[[File:2010- Decreasing renewable energy costs versus deployment.svg|thumb|upright=1.3|A log–log plot condensing information that spans more than one order of magnitude along both axes]]
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 ===
{{Main|Semi-log plot}}
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) to avoid the singularity for zero input (''y'' = 0), and so produce symmetric log plots:<ref name="Webber2012">{{cite journal | last=Webber | first=J Beau W | title=A bi-symmetric log transformation for wide-range data | journal=Measurement Science and Technology | publisher=IOP Publishing | volume=24 | issue=2 | date=2012-12-21 | issn=0957-0233 | doi=10.1088/0957-0233/24/2/027001 | page=027001| s2cid=12007380 | url=https://kar.kent.ac.uk/32810/2/2012_Bi-symmetric-log-transformation_v5.pdf }}</ref><ref name="Matplotlib 3.4.2 documentation 2021">{{cite web | title=Symlog Demo | website=Matplotlib 3.4.2 documentation | date=2021-05-08 | url=https://matplotlib.org/stable/gallery/scales/symlog_demo.html | access-date=2021-06-22}}</ref>
:<math>Y=\sgn(y)\cdot\log_{10}(1+|y/C|)</math>
for a constant ''C''=1/ln(10).

== Logarithmic units ==
A '''logarithmic unit''' is a [[Units of measurement|unit]] that can be used to express a quantity ([[Physical 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 information]] and [[information entropy]] ([[nat (unit)|nat]], [[shannon (unit)|shannon]], [[ban (information)|ban]]) and of [[signal level]] ([[decibel]], bel, [[neper]]).  [[Frequency level]]s or logarithmic frequency quantities have various units are used in electronics ([[Decade (log scale)|decade]], [[octave (electronics)|octave]]) and for music pitch [[Interval (music)|interval]]s ([[octave]], [[semitone]], [[Cent (music)|cent]], etc.).  Other logarithmic scale units include the [[Richter magnitude scale]] point.

In addition, several industrial measures are logarithmic, such as standard values for [[Logarithmic_resistor_ladder|resistors]], the [[American wire gauge]], the [[Birmingham gauge]] used for wire and needles, and so on.

=== Units of information ===
* [[bit]], [[byte]]
* [[hartley (unit)|hartley]]
* [[nat (unit)|nat]]
* [[shannon (unit)|shannon]]

=== Units of level or level difference ===
{{further|Level (logarithmic quantity)}}
* [[bel (unit)|bel]], [[decibel]]
* [[neper]]

==== Units of frequency level ====
* [[Decade (log scale)|decade]], [[One-third octave|decidecade]], [[savart]]
* [[octave]], [[Whole tone|tone]], [[semitone]], [[Cent (music)|cent]]

=== Table of examples ===

{| class="wikitable"
|-
! Unit
! Base of logarithm
! Underlying quantity
! Interpretation
|-
| [[bit]]
| {{math|2}}
| number of possible messages
| [[Quantities of information|quantity of information]]
|-
| [[byte]]
| {{math|2<sup>8</sup> {{=}} 256}}
| number of possible messages
| [[Quantities of information|quantity of information]]
|-
| [[decibel]]
| {{math|10<sup>(1/10)</sup> ≈ 1.259}}
| any [[Power, root-power, and field quantities|power quantity]] ([[sound power]], for example)
| [[sound power level]] (for example)
|-
| [[decibel]]
| {{math|10<sup>(1/20)</sup> ≈ 1.122}}
| any [[Power, root-power, and field quantities|root-power quantity]] ([[sound pressure]], for example)
| [[sound pressure level]] (for example)
|-
| [[semitone]]
| {{math|2<sup>(1/12)</sup> ≈ 1.059}}
| [[frequency]] of [[sound]]
| [[pitch interval]]
|}

The two definitions of a decibel are equivalent, because a ratio of [[Power, root-power, and field quantities|power quantities]] is equal to the square of the corresponding ratio of [[Power, root-power, and field quantities|root-power quantities]].{{citation needed|reason=The two definitions of the logarithmic unit or of the quantity are not equivalent.  The same numeric value is obtained in a linear system with a constant impedance and the choice of units, but they are inherently different units despite having the same name.|date=December 2019}}<ref>[https://repository.oceanbestpractices.org/bitstream/handle/11329/2340/A-Century-of-Sonar.pdf?sequence=1 Ainslie, M. A. (2015). A Century of Sonar: Planetary Oceanography, Underwater Noise Monitoring, and the Terminology of Underwater Sound.]</ref>

== See also ==
{{Portal|Mathematics}}
* [[Alexander Graham Bell]]
* [[Bode plot]]
* [[Geometric mean]] (arithmetic mean in logscale)
* [[John Napier]]
* [[Level (logarithmic quantity)]]
* [[Log–log plot]]
* [[Logarithm]]
* [[Logarithmic mean]]
* [[Log semiring]]
* [[Preferred number]]
* [[Semi-log plot]]

=== Scale ===
* [[Order of magnitude]]

=== Applications ===
* [[Entropy]]
* [[Entropy (information theory)]]
* [[pH]]
* [[Richter magnitude scale]]

== References ==
{{Reflist}}

== Further reading ==
* {{cite journal |author-first1=Stanislas |author-last1=Dehaene |author-first2=Véronique |author-last2=Izard |author-first3=Elizabeth |author-last3=Spelke |author-link3=Elizabeth Spelke |author-first4=Pierre |author-last4=Pica |author-link4=Pierre Pica |date=2008 |title=Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures |journal=Science |volume=320 |issue=5880 |doi=10.1126/science.1156540 |pmid=18511690 |pmc=2610411 |pages=1217–20 |bibcode=2008Sci...320.1217D}}
* {{cite journal |author-first1=Karl |author-last1=Tuffentsammer |author-first2=P. |author-last2=Schumacher |title=Normzahlen – die einstellige Logarithmentafel des Ingenieurs |language=de |trans-title=Preferred numbers - the engineer's single-digit logarithm table |journal=Werkstattechnik und Maschinenbau |volume=43 |number=4 |date=1953 |page=156}}
* {{cite journal |author-last=Tuffentsammer |author-first=Karl |title=Das Dezilog, eine Brücke zwischen Logarithmen, Dezibel, Neper und Normzahlen |language=de |trans-title=The decilog, a bridge between logarithms, decibel, neper and preferred numbers |journal=VDI-Zeitschrift |volume=98 |date=1956 |pages=267–274}}
* {{cite book |author-first=Clemens |author-last=Ries |title=Normung nach Normzahlen |language=de |trans-title=Standardization by preferred numbers |publisher=Duncker & Humblot Verlag |location=Berlin, Germany |date=1962 |edition=1 |isbn=978-3-42801242-8}} (135 pages)
* {{cite book |title=Logarithmen, Normzahlen, Dezibel, Neper, Phon - natürlich verwandt! |language=de |trans-title=Logarithms, preferred numbers, decibel, neper, phon - naturally related! |author-first=Eugen |author-last=Paulin |date=2007-09-01 |url=http://www.rechenschieber.org/Normzahlen.pdf |access-date=2016-12-18 |url-status=live |archive-url=https://web.archive.org/web/20161218223050/http://www.rechenschieber.org/Normzahlen.pdf |archive-date=2016-12-18}}

== External links ==
{{Commonscat}}
* {{cite web|url=https://www.gnu.org/software/emacs/manual/html_node/calc/Logarithmic-Units.html |title=GNU Emacs Calc Manual: Logarithmic Units |website=Gnu.org |access-date=2016-11-23}}
* [https://sites.google.com/site/nonnewtoniancalculus/ Non-Newtonian calculus website]

{{DEFAULTSORT:Logarithmic Scale}}
[[Category:Logarithmic scales of measurement| ]]
[[Category:Non-Newtonian calculus]]