Editing Logarithm (section)

==Applications==
[[File:NautilusCutawayLogarithmicSpiral.jpg|thumb|A [[nautilus]] shell displaying a logarithmic spiral|alt=A photograph of a nautilus' shell.]]
Logarithms have many applications inside and outside mathematics. Some of these occurrences are related to the notion of [[scale invariance]]. For example, each chamber of the shell of a [[nautilus]] is an approximate copy of the next one, scaled by a constant factor. This gives rise to a [[logarithmic spiral]].<ref>{{Harvard citations
 |last1=Maor
 |year=2009
 |nb=yes
 |loc=p. 135
 }}</ref> [[Benford's law]] on the distribution of leading digits can also be explained by scale invariance.<ref>{{Citation | last1=Frey | first1=Bruce | title=Statistics hacks | publisher=[[O'Reilly Media|O'Reilly]]|location=Sebastopol, CA| series=Hacks Series |url={{google books |plainurl=y |id=HOPyiNb9UqwC|page=275}}| isbn=978-0-596-10164-0 | year=2006}}, chapter 6, section 64</ref> Logarithms are also linked to [[self-similarity]]. For example, logarithms appear in the analysis of algorithms that solve a problem by dividing it into two similar smaller problems and patching their solutions.<ref>{{Citation | last1=Ricciardi | first1=Luigi M. | title=Lectures in applied mathematics and informatics | url={{google books |plainurl=y |id=Cw4NAQAAIAAJ}} | publisher=Manchester University Press | location=Manchester | isbn=978-0-7190-2671-3 | year=1990}}, p.&nbsp;21, section 1.3.2</ref> The dimensions of self-similar geometric shapes, that is, shapes whose parts resemble the overall picture are also based on logarithms. [[Logarithmic scale]]s are useful for quantifying the relative change of a value as opposed to its absolute difference. Moreover, because the logarithmic function {{math|log(''x'')}} grows very slowly for large {{mvar|x}}, logarithmic scales are used to compress large-scale scientific data. Logarithms also occur in numerous scientific formulas, such as the [[Tsiolkovsky rocket equation]], the [[Fenske equation]], or the [[Nernst equation]].

===Logarithmic scale===
{{Main|Logarithmic scale}}
[[File:Germany Hyperinflation.svg|A logarithmic chart depicting the value of one [[German gold mark|Goldmark]] in [[German Papiermark|Papiermarks]] during the [[Inflation in the Weimar Republic|German hyperinflation in the 1920s]]|right|thumb|alt=A graph of the value of one mark over time. The line showing its value is increasing very quickly, even with logarithmic scale.]]
Scientific quantities are often expressed as logarithms of other quantities, using a ''logarithmic scale''. For example, the [[decibel]] is a [[unit of measurement]] associated with [[logarithmic-scale]] [[level quantity|quantities]]. It is based on the common logarithm of [[ratio]]s—10&nbsp;times the common logarithm of a [[power (physics)|power]] ratio or 20&nbsp;times the common logarithm of a [[voltage]] ratio. It is used to quantify the attenuation or amplification of electrical signals,<ref>{{cite book|contribution=7.5.1 Decibel (dB)|title=Power Quality|first=C.|last=Sankaran|publisher=Taylor & Francis|year=2001|isbn=9780849310409|quote=The decibel is used to express the ratio between two quantities. The quantities may be voltage, current, or power.}}</ref> to describe power levels of sounds in [[acoustics]],<ref>{{Citation|last1=Maling|first1=George C.|editor1-last=Rossing|editor1-first=Thomas D.|title=Springer handbook of acoustics|publisher=[[Springer-Verlag]]|location=Berlin, New York|isbn=978-0-387-30446-5|year=2007|chapter=Noise}}, section 23.0.2</ref> and the [[absorbance]] of light in the fields of [[spectrometer|spectrometry]] and [[optics]]. The [[signal-to-noise ratio]] describing the amount of unwanted [[noise (electronic)|noise]] in relation to a (meaningful) [[signal (information theory)|signal]] is also measured in decibels.<ref>{{Citation | last1=Tashev | first1=Ivan Jelev | title=Sound Capture and Processing: Practical Approaches | publisher=[[John Wiley & Sons]] | location=New York | isbn=978-0-470-31983-3 | year=2009|url={{google books |plainurl=y |id=plll9smnbOIC|page=48}}|page=98}}</ref> In a similar vein, the [[peak signal-to-noise ratio]] is commonly used to assess the quality of sound and [[image compression]] methods using the logarithm.<ref>{{Citation | last1=Chui | first1=C.K. | title=Wavelets: a mathematical tool for signal processing | publisher=[[Society for Industrial and Applied Mathematics]] | location=Philadelphia | series=SIAM monographs on mathematical modeling and computation | isbn=978-0-89871-384-8 | year=1997|url={{google books |plainurl=y |id=N06Gu433PawC|page=180}}}}</ref>

The strength of an earthquake is measured by taking the common logarithm of the energy emitted at the quake. This is used in the [[moment magnitude scale]] or the [[Richter magnitude scale]]. For example, a 5.0 earthquake releases 32&nbsp;times {{math|(10<sup>1.5</sup>)}} and a 6.0 releases 1000&nbsp;times {{math|(10<sup>3</sup>)}} the energy of a 4.0.<ref>{{Citation|last1=Crauder|first1=Bruce|last2=Evans|first2=Benny|last3=Noell|first3=Alan|title=Functions and Change: A Modeling Approach to College Algebra|publisher=Cengage Learning|location=Boston|edition=4th|isbn=978-0-547-15669-9|year=2008}}, section 4.4.</ref> [[Apparent magnitude]] measures the brightness of stars logarithmically.<ref>{{Citation|last1=Bradt|first1=Hale|title=Astronomy methods: a physical approach to astronomical observations|publisher=[[Cambridge University Press]]|series=Cambridge Planetary Science|isbn=978-0-521-53551-9|year=2004}}, section 8.3, p.&nbsp;231</ref> In [[chemistry]] the negative of the decimal logarithm, the decimal '''{{vanchor|cologarithm}}''', is indicated by the letter p.<ref name="Jens">{{cite journal|author=Nørby, Jens|year=2000|title=The origin and the meaning of the little p in pH|journal=Trends in Biochemical Sciences|volume=25|issue=1|pages=36–37|doi=10.1016/S0968-0004(99)01517-0|pmid=10637613}}</ref> For instance, [[pH]] is the decimal cologarithm of the [[Activity (chemistry)|activity]] of [[hydronium]] ions (the form [[hydrogen]] [[ion]]s {{H+}} take in water).<ref>{{Citation|author=IUPAC|title=Compendium of Chemical Terminology ("Gold Book")|edition=2nd|editor=A. D. McNaught, A. Wilkinson|publisher=Blackwell Scientific Publications| location=Oxford| year=1997| url=http://goldbook.iupac.org/P04524.html|isbn=978-0-9678550-9-7|doi=10.1351/goldbook|author-link=IUPAC|doi-access=free}}</ref> The activity of hydronium ions in neutral water is 10<sup>−7</sup>&nbsp;[[molar concentration|mol·L<sup>−1</sup>]], hence a pH of 7. Vinegar typically has a pH of about 3. The difference of 4 corresponds to a ratio of 10<sup>4</sup> of the activity, that is, vinegar's hydronium ion activity is about {{math|10<sup>−3</sup> mol·L<sup>−1</sup>}}.

[[Semi-log plot|Semilog]] (log–linear) graphs use the logarithmic scale concept for visualization: one axis, typically the vertical one, is scaled logarithmically. For example, the chart at the right compresses the steep increase from 1&nbsp;million to 1&nbsp;trillion to the same space (on the vertical axis) as the increase from 1 to 1&nbsp;million. In such graphs, [[exponential function]]s of the form {{math|1=''f''(''x'') = ''a'' · ''b''{{i sup|''x''}}}} appear as straight lines with [[slope]] equal to the logarithm of {{mvar|b}}. [[Log-log plot|Log-log]] graphs scale both axes logarithmically, which causes functions of the form {{math|1=''f''(''x'') = ''a'' · ''x''{{i sup|''k''}}}} to be depicted as straight lines with slope equal to the exponent&nbsp;{{mvar|k}}. This is applied in visualizing and analyzing [[power law]]s.<ref>{{Citation|last1=Bird|first1=J.O.|title=Newnes engineering mathematics pocket book  |publisher=Newnes|location=Oxford|edition=3rd|isbn=978-0-7506-4992-6|year=2001}}, section 34</ref>

===Psychology===
Logarithms occur in several laws describing [[human perception]]:<ref>{{Citation | last1=Goldstein | first1=E. Bruce | title=Encyclopedia of Perception | url={{google books |plainurl=y |id=Y4TOEN4f5ZMC}} | publisher=Sage | location=Thousand Oaks, CA | isbn=978-1-4129-4081-8 | year=2009}}, pp.&nbsp;355–56</ref><ref>{{Citation | last1=Matthews | first1=Gerald | title=Human Performance: Cognition, Stress, and Individual Differences | url={{google books |plainurl=y |id=0XrpulSM1HUC}} | publisher=Psychology Press | location=Hove | isbn=978-0-415-04406-6 | year=2000}}, p.&nbsp;48</ref> [[Hick's law]] proposes a logarithmic relation between the time individuals take to choose an alternative and the number of choices they have.<ref>{{Citation|last1=Welford|first1=A.T.|title=Fundamentals of skill|publisher=Methuen|location=London|isbn=978-0-416-03000-6 |oclc=219156|year=1968}}, p.&nbsp;61</ref> [[Fitts's law]] predicts that the time required to rapidly move to a target area is a logarithmic function of the ratio between the distance to a target and the size of the target.<ref>{{Citation|author=Paul M. Fitts|date=June 1954|title=The information capacity of the human motor system in controlling the amplitude of movement|journal=Journal of Experimental Psychology|volume=47|issue=6|pages=381–91 | pmid=13174710 | doi =10.1037/h0055392 |s2cid=501599}}, reprinted in {{Citation|journal=Journal of Experimental Psychology: General|volume=121|issue=3|pages=262–69|year=1992 | pmid=1402698 | url=http://sing.stanford.edu/cs303-sp10/papers/1954-Fitts.pdf | access-date=30 March 2011 |title=The information capacity of the human motor system in controlling the amplitude of movement|author=Paul M. Fitts|doi=10.1037/0096-3445.121.3.262}}</ref> In [[psychophysics]], the [[Weber–Fechner law]] proposes a logarithmic relationship between [[stimulus (psychology)|stimulus]] and [[sensation (psychology)|sensation]] such as the actual vs. the perceived weight of an item a person is carrying.<ref>{{Citation | last1=Banerjee | first1=J.C. | title=Encyclopaedic dictionary of psychological terms | publisher=M.D. Publications | location=New Delhi | isbn=978-81-85880-28-0  | oclc=33860167 | year=1994|url={{google books |plainurl=y |id=Pwl5U2q5hfcC|page=306}} |page=304}}</ref> (This "law", however, is less realistic than more recent models, such as [[Stevens's power law]].<ref>{{Citation|last1=Nadel|first1=Lynn|author1-link=Lynn Nadel|title=Encyclopedia of cognitive science|publisher=[[John Wiley & Sons]]|location=New York|isbn=978-0-470-01619-0|year=2005}}, lemmas ''Psychophysics'' and ''Perception: Overview''</ref>)

Psychological studies found that individuals with little mathematics education tend to estimate quantities logarithmically, that is, they position a number on an unmarked line according to its logarithm, so that 10 is positioned as close to 100 as 100 is to 1000. Increasing education shifts this to a linear estimate (positioning 1000 10 times as far away) in some circumstances, while logarithms are used when the numbers to be plotted are difficult to plot linearly.<ref>
{{Citation|doi=10.1111/1467-9280.02438|last1=Siegler|first1=Robert S.|last2=Opfer|first2=John E.|title=The Development of Numerical Estimation. Evidence for Multiple Representations of Numerical Quantity|volume=14|issue=3|pages=237–43|year=2003|journal=Psychological Science|url=http://www.psy.cmu.edu/~siegler/sieglerbooth-cd04.pdf|pmid=12741747|citeseerx=10.1.1.727.3696|s2cid=9583202|access-date=7 January 2011|archive-url=https://web.archive.org/web/20110517002232/http://www.psy.cmu.edu/~siegler/sieglerbooth-cd04.pdf|archive-date=17 May 2011|url-status=dead}}
</ref><ref>{{Citation|last1=Dehaene| first1=Stanislas|last2=Izard|first2=Véronique |last3=Spelke| first3=Elizabeth|last4=Pica| first4=Pierre| title=Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures|volume=320|issue=5880|pages=1217–20|doi=10.1126/science.1156540|pmc=2610411|pmid=18511690| year=2008|journal=Science|bibcode=2008Sci...320.1217D| citeseerx=10.1.1.362.2390}}</ref>

===Probability theory and statistics===
[[File:PDF-log normal distributions.svg|thumb|right|alt=Three asymmetric PDF curves|Three [[probability density function]]s (PDF) of random variables with log-normal distributions. The location parameter&nbsp;{{math|μ}}, which is zero for all three of the PDFs shown, is the mean of the logarithm of the random variable, not the mean of the variable itself.]]
[[File:Benfords law illustrated by world's countries population.svg|Distribution of first digits (in %, red bars) in the [[List of countries by population|population of the 237 countries]] of the world. Black dots indicate the distribution predicted by Benford's law.|thumb|right|alt=A bar chart and a superimposed second chart. The two differ slightly, but both decrease in a similar fashion.]]

Logarithms arise in [[probability theory]]: the [[law of large numbers]] dictates that, for a [[fair coin]], as the number of coin-tosses increases to infinity, the observed proportion of heads [[binomial distribution|approaches one-half]]. The fluctuations of this proportion about one-half are described by the [[law of the iterated logarithm]].<ref>{{Citation | last1=Breiman | first1=Leo | title=Probability | publisher=[[Society for Industrial and Applied Mathematics]] | location=Philadelphia | series=Classics in applied mathematics | isbn=978-0-89871-296-4 | year=1992}}, section 12.9</ref>

Logarithms also occur in [[log-normal distribution]]s. When the logarithm of a [[random variable]] has a [[normal distribution]], the variable is said to have a log-normal distribution.<ref>{{Citation|last1=Aitchison|first1=J.|last2=Brown|first2=J.A.C.|title=The lognormal distribution|publisher=[[Cambridge University Press]]|isbn=978-0-521-04011-2 |oclc=301100935|year=1969}}</ref> Log-normal distributions are encountered in many fields, wherever a variable is formed as the product of many independent positive random variables, for example in the study of turbulence.<ref>
{{Citation
 | title = An introduction to turbulent flow
 | author = Jean Mathieu and Julian Scott
 | publisher = Cambridge University Press
 | year = 2000
 | isbn = 978-0-521-77538-0
 | page = 50
 | url = {{google books |plainurl=y |id=nVA53NEAx64C|page=50}}
 }}</ref>

Logarithms are used for [[maximum-likelihood estimation]] of parametric [[statistical model]]s. For such a model, the [[likelihood function]] depends on at least one [[parametric model|parameter]] that must be estimated. A maximum of the likelihood function occurs at the same parameter-value as a maximum of the logarithm of the likelihood (the "''log&nbsp;likelihood''"), because the logarithm is an increasing function. The log-likelihood is easier to maximize, especially for the multiplied likelihoods for [[independence (probability)|independent]] random variables.<ref>{{Citation|last1=Rose|first1=Colin|last2=Smith|first2=Murray D.|title=Mathematical statistics with Mathematica|publisher=[[Springer-Verlag]]|location=Berlin, New York|series=Springer texts in statistics|isbn=978-0-387-95234-5|year=2002}}, section 11.3</ref>

[[Benford's law]] describes the occurrence of digits in many [[data set]]s, such as heights of buildings. According to Benford's law, the probability that the first decimal-digit of an item in the data sample is {{Mvar|d}} (from 1 to 9) equals {{math|log<sub>10</sub>&thinsp;(''d'' + 1) − log<sub>10</sub>&thinsp;(''d'')}}, ''regardless'' of the unit of measurement.<ref>{{Citation|last1=Tabachnikov|first1=Serge|author-link1=Sergei Tabachnikov|title=Geometry and Billiards|publisher=[[American Mathematical Society]]|location=Providence, RI|isbn=978-0-8218-3919-5|year=2005|pages=36–40}}, section 2.1</ref> Thus, about 30% of the data can be expected to have 1 as first digit, 18% start with 2, etc. Auditors examine deviations from Benford's law to detect fraudulent accounting.<ref>{{citation |title=The Effective Use of Benford's Law in Detecting Fraud in Accounting Data |first1=Cindy |last1=Durtschi |first2=William |last2=Hillison |first3=Carl |last3=Pacini |url=http://faculty.usfsp.edu/gkearns/Articles_Fraud/Benford%20Analysis%20Article.pdf |volume=V |pages=17–34 |year=2004 |journal=Journal of Forensic Accounting |archive-url=https://web.archive.org/web/20170829062510/http://faculty.usfsp.edu/gkearns/Articles_Fraud/Benford%20Analysis%20Article.pdf |archive-date=29 August 2017 |access-date=28 May 2018}}</ref>

The [[logarithm transformation]] is a type of [[data transformation (statistics)|data transformation]] used to bring the empirical distribution closer to the assumed one.

===Computational complexity===
[[Analysis of algorithms]] is a branch of [[computer science]] that studies the [[time complexity|performance]] of [[algorithm]]s (computer programs solving a certain problem).<ref name=Wegener>{{Citation|last1=Wegener|first1=Ingo| title=Complexity theory: exploring the limits of efficient algorithms|publisher=[[Springer-Verlag]]|location=Berlin, New York|isbn=978-3-540-21045-0|year=2005}}, pp. 1–2</ref> Logarithms are valuable for describing algorithms that [[Divide and conquer algorithm|divide a problem]] into smaller ones, and join the solutions of the subproblems.<ref>{{Citation|last1=Harel|first1=David|last2=Feldman|first2=Yishai A.|title=Algorithmics: the spirit of computing|location=New York|publisher=[[Addison-Wesley]]|isbn=978-0-321-11784-7|year=2004}}, p.&nbsp;143</ref>

For example, to find a number in a sorted list, the [[binary search algorithm]] checks the middle entry and proceeds with the half before or after the middle entry if the number is still not found. This algorithm requires, on average, {{math|log<sub>2</sub>&thinsp;(''N'')}} comparisons, where {{mvar|N}} is the list's length.<ref>{{citation  | last = Knuth  | first = Donald  | author-link = Donald Knuth  | title = The Art of Computer Programming  | publisher = Addison-Wesley  |location=Reading, MA | year=  1998| isbn = 978-0-201-89685-5 | title-link = The Art of Computer Programming  }}, section 6.2.1, pp. 409–26</ref> Similarly, the [[merge sort]] algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results. Merge sort algorithms typically require a time [[big O notation|approximately proportional to]] {{math|''N'' · log(''N'')}}.<ref>{{Harvard citations|last = Knuth  | first = Donald|year=1998|loc=section 5.2.4, pp. 158–68|nb=yes}}</ref> The base of the logarithm is not specified here, because the result only changes by a constant factor when another base is used. A constant factor is usually disregarded in the analysis of algorithms under the standard [[uniform cost model]].<ref name=Wegener20>{{Citation|last1=Wegener|first1=Ingo| title=Complexity theory: exploring the limits of efficient algorithms|publisher=[[Springer-Verlag]]|location=Berlin, New York|isbn=978-3-540-21045-0|year=2005|page=20}}</ref>

A function&nbsp;{{math|''f''(''x'')}} is said to [[Logarithmic growth|grow logarithmically]] if {{math|''f''(''x'')}} is (exactly or approximately) proportional to the logarithm of {{mvar|x}}. (Biological descriptions of organism growth, however, use this term for an exponential function.<ref>{{Citation|last1=Mohr|first1=Hans|last2=Schopfer|first2=Peter|title=Plant physiology|publisher=Springer-Verlag|location=Berlin, New York|isbn=978-3-540-58016-4|year=1995|url-access=registration|url=https://archive.org/details/plantphysiology0000mohr}}, chapter 19, p.&nbsp;298</ref>) For example, any [[natural number]]&nbsp;{{mvar|N}} can be represented in [[Binary numeral system|binary form]] in no more than {{math|log<sub>2</sub>&thinsp;''N'' + 1}}&nbsp;[[bit]]s. In other words, the amount of [[memory (computing)|memory]] needed to store {{mvar|N}} grows logarithmically with {{mvar|N}}.

===Entropy and chaos===
[[File:Chaotic Bunimovich stadium.svg|right|thumb|[[Dynamical billiards|Billiards]] on an oval [[billiard table]]. Two particles, starting at the center with an angle differing by one degree, take paths that diverge chaotically because of [[reflection (physics)|reflections]] at the boundary.|alt=An oval shape with the trajectories of two particles.]]

[[Entropy]] is broadly a measure of the disorder of some system. In [[statistical thermodynamics]], the entropy&nbsp;{{mvar|S}} of some physical system is defined as
<math display="block"> S = - k \sum_i p_i \ln(p_i).\, </math>
The sum is over all possible states&nbsp;{{Mvar|i}} of the system in question, such as the positions of gas particles in a container. Moreover, {{math|''p''<sub>''i''</sub>}} is the probability that the state&nbsp;{{Mvar|i}} is attained and {{mvar|k}} is the [[Boltzmann constant]]. Similarly, [[entropy (information theory)|entropy in information theory]] measures the quantity of information. If a message recipient may expect any one of {{mvar|N}} possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as {{math|log<sub>2</sub>&thinsp;''N''}} bits.<ref>{{Citation|last1=Eco|first1=Umberto|author1-link=Umberto Eco|title=The open work  |publisher=[[Harvard University Press]]|isbn=978-0-674-63976-8|year=1989}}, section III.I</ref>

[[Lyapunov exponent]]s use logarithms to gauge the degree of chaoticity of a [[dynamical system]]. For example, for a particle moving on an oval billiard table, even small changes of the initial conditions result in very different paths of the particle. Such systems are [[chaos theory|chaotic]] in a [[Deterministic system|deterministic]] way, because small measurement errors of the initial state predictably lead to largely different final states.<ref>{{Citation | last1=Sprott | first1=Julien Clinton | title=Elegant Chaos: Algebraically Simple Chaotic Flows | journal=Elegant Chaos: Algebraically Simple Chaotic Flows. Edited by Sprott Julien Clinton. Published by World Scientific Publishing Co. Pte. Ltd | url={{google books |plainurl=y |id=buILBDre9S4C}} | publisher=[[World Scientific]] |location=New Jersey|isbn=978-981-283-881-0| year=2010| bibcode=2010ecas.book.....S | doi=10.1142/7183 }}, section 1.9</ref> At least one Lyapunov exponent of a deterministically chaotic system is positive.

===Fractals===
[[File:Sierpinski dimension.svg|The Sierpinski triangle (at the right) is constructed by repeatedly replacing [[equilateral triangle]]s by three smaller ones.|thumb|alt=Parts of a triangle are removed in an iterated way.]]

Logarithms occur in definitions of the [[fractal dimension|dimension]] of [[fractal]]s.<ref>{{Citation|last1=Helmberg|first1=Gilbert|title=Getting acquainted with fractals|publisher=Walter de Gruyter|series=De Gruyter Textbook|location=Berlin, New York|isbn=978-3-11-019092-2|year=2007}}</ref> Fractals are geometric objects that are self-similar in the sense that small parts reproduce, at least roughly, the entire global structure. The [[Sierpinski triangle]] (pictured) can be covered by three copies of itself, each having sides half the original length. This makes the [[Hausdorff dimension]] of this structure {{math|1=ln(3)/ln(2) ≈ 1.58}}. Another logarithm-based notion of dimension is obtained by [[box-counting dimension|counting the number of boxes]] needed to cover the fractal in question.

===Music===
{{multiple image
| direction         = vertical
| width             = 350
| footer            = Four different octaves shown on a linear scale, then shown on a logarithmic scale (as the ear hears them)
| image1            = 4Octaves.and.Frequencies.svg
| alt1              = Four different octaves shown on a linear scale.
| image2            = 4Octaves.and.Frequencies.Ears.svg
| alt2              = Four different octaves shown on a logarithmic scale
}}

Logarithms are related to musical tones and [[interval (music)|intervals]]. In [[equal temperament]] tunings, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or [[pitch (music)|pitch]], of the individual tones. In the [[12-tone equal temperament]] tuning common in modern Western music, each [[octave]] (doubling of frequency) is broken into twelve equally spaced intervals called [[semitone]]s. For example, if the [[a (musical note)|note&nbsp;''A'']] has a frequency of 440&nbsp;[[Hertz|Hz]] then the note [[B♭ (musical note)|''B-flat'']] has a frequency of 466&nbsp;Hz. The interval between ''A'' and ''B-flat'' is a [[semitone]], as is the one between ''B-flat'' and [[b (musical note)|''B'']] (frequency 493&nbsp;Hz). Accordingly, the frequency ratios agree:
<math display="block">\frac{466}{440} \approx \frac{493}{466} \approx 1.059 \approx \sqrt[12]2.</math>

Intervals between arbitrary pitches can be measured in octaves by taking the {{Nowrap|base-{{math|2}}}} logarithm of the [[frequency]] ratio, can be measured in equally tempered semitones by taking the {{Nowrap|base-{{math|2<sup>1/12</sup>}}}} logarithm ({{math|12}} times the {{Nowrap|base-{{math|2}}}} logarithm), or can be measured in [[cent (music)|cents]], hundredths of a semitone, by taking the {{Nowrap|base-{{math|2<sup>1/1200</sup>}}}} logarithm ({{math|1200}} times the {{Nowrap|base-{{math|2}}}} logarithm). The latter is used for finer encoding, as it is needed for finer measurements or non-equal temperaments.<ref>{{Citation|last1=Wright|first1=David|title=Mathematics and music|location=Providence, RI|publisher=AMS Bookstore|isbn=978-0-8218-4873-9|year=2009}}, chapter 5</ref>

{| class="wikitable" style="text-align:center;"
|-
! Interval<br /> <span style="font-weight: normal">(the two tones are played<br> at the same time)</span>
| [[72 tone equal temperament|1/12 tone]]<br> {{audio|1_step_in_72-et_on_C.mid|play}}
| [[Semitone]]<br> {{audio|help=no|Minor_second_on_C.mid|play}}
| [[Just major third]]<br> {{audio|help=no|Just_major_third_on_C.mid|play}}
| [[Major third]]<br> {{audio|help=no|Major_third_on_C.mid|play}}
| [[Tritone]]<br> {{audio|help=no|Tritone_on_C.mid|play}}
| [[Octave]]<br> {{audio|help=no|Perfect_octave_on_C.mid|play}}
|-
! '''Frequency ratio'''<br> <math>r</math>
| <math>2^{\frac 1 {72}} \approx 1.0097</math>
| <math>2^{\frac 1 {12}} \approx 1.0595</math>
| <math>\tfrac 5 4 = 1.25</math>
| <math>\begin{align} 2^{\frac 4 {12}} & = \sqrt[3] 2 \\ & \approx 1.2599 \end{align} </math>
| <math>\begin{align} 2^{\frac 6 {12}} & = \sqrt 2 \\ & \approx 1.4142 \end{align} </math>
| <math> 2^{\frac {12} {12}} = 2 </math>
|-
! '''Number of semitones'''<br /><math>12 \log_2 r</math>
| <math>\tfrac 1 6</math>
| <math>1</math>
| <math>\approx 3.8631</math>
| <math>4</math>
| <math>6</math>
| <math>12</math>
|-
! '''Number of cents'''<br /><math>1200 \log_2 r</math>
| <math>16 \tfrac 2 3</math>
| <math>100</math>
| <math>\approx 386.31</math>
| <math>400</math>
| <math>600</math>
| <math>1200</math>
|}

===Number theory===
[[Natural logarithm]]s are closely linked to [[prime-counting function|counting prime numbers]] (2, 3, 5, 7, 11, ...), an important topic in [[number theory]]. For any [[integer]]&nbsp;{{mvar|x}}, the quantity of [[prime number]]s less than or equal to {{mvar|x}} is denoted {{math|[[prime-counting function|{{pi}}(''x'')]]}}. The [[prime number theorem]] asserts that {{math|{{pi}}(''x'')}} is approximately given by
<math display="block">\frac{x}{\ln(x)},</math>
in the sense that the ratio of {{math|{{pi}}(''x'')}} and that fraction approaches 1 when {{mvar|x}} tends to infinity.<ref>{{Citation|last1=Bateman|first1=P.T.|last2=Diamond|first2=Harold G.|title=Analytic number theory: an introductory course|publisher=[[World Scientific]]|location=New Jersey|isbn=978-981-256-080-3 |oclc=492669517|year=2004}}, theorem 4.1</ref> As a consequence, the probability that a randomly chosen number between 1 and {{mvar|x}} is prime is inversely [[proportionality (mathematics)|proportional]] to the number of decimal digits of {{mvar|x}}. A far better estimate of {{math|{{pi}}(''x'')}} is given by the [[logarithmic integral function|offset logarithmic integral]] function {{math|Li(''x'')}}, defined by
<math display="block"> \mathrm{Li}(x) = \int_2^x \frac1{\ln(t)} \,dt.  </math>
The [[Riemann hypothesis]], one of the oldest open mathematical [[conjecture]]s, can be stated in terms of comparing {{math|{{pi}}(''x'')}} and {{math|Li(''x'')}}.<ref>{{Harvard citations|last1=Bateman|first1=P. T.|last2=Diamond|year=2004|nb=yes |loc=Theorem 8.15}}</ref> The [[Erdős–Kac theorem]] describing the number of distinct [[prime factor]]s also involves the [[natural logarithm]].

The logarithm of ''n'' [[factorial]], {{math|1=''n''! = 1 · 2 · ... · ''n''}}, is given by
<math display="block"> \ln (n!) = \ln (1) + \ln (2) + \cdots + \ln (n).</math>
This can be used to obtain [[Stirling's formula]], an approximation of {{math|''n''!}} for large {{mvar|n}}.<ref>{{Citation|last1=Slomson|first1=Alan B.|title=An introduction to combinatorics|publisher=[[CRC Press]]|location=London|isbn=978-0-412-35370-3|year=1991}}, chapter 4</ref>