Editing Logarithm (section)

===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.