Pengguna:Dedhert.Jr/Uji halaman 01/16: Perbedaan antara revisi

: <math> ^b\!\log(xy) = \, ^b\!\log x + \, ^b\!\log y,</math>

 

 

[[Skala logaritma]] mengurangi jumlah luas ke lingkup yang lebih kecil. Misalnya, [[desibel]] (dB) adalah [[Satuan pengukuran|satuan]] yang digunakan untuk menyatakan [[Tingkat (kuantitas logaritmik)|rasio sebagai logaritma]], sebagian besar untuk kekuatan sinyal dan amplitudo (contoh umumnya pada [[tekanan suara]]). Dalam kimia, [[pH]] mengukur [[Asam|keasaman]] dari [[larutan berair]] melalui logaritma. Logaritma biasa dalam [[rumus]] ilmiah, dan dalam pengukuran [[Teori kompleksitas komputasi|kompleksitas algoritma]] dan objek geometris yang disebut [[fraktal]]. Logaritma juga membantu untuk menjelaskan [[frekuensi]] rasio [[Interval (musik)|interval musik]], muncul dalam rumus yang menghitung [[bilangan prima]] atau [[Hampiran Stirling|hampiran]] [[faktorial]], memberikan gambaran dalam [[psikofisika]], dan dapat membantu perhitungan [[akuntansi forensik]].

:<math>\log(\cos \theta + i\sin \theta) = i\theta</math>.

 

[[Berkas:Logarithms_Britannica_1797.png|ka|jmpl|360x360px|Penjelasan logaritma dalam ''[[Encyclopædia Britannica]]'' pada tahun 1797.]]

Dengan menyederhanakan perhitungan yang rumit sebelum adanya mesin hitung komputer, logaritma berkontribusi pada kemajuan pengetahuan, khususnya [[astronomi]]. Logaritma sangat penting terhadap kemajuan dalam [[survei]], [[navigasi benda langit]], dan cabang lainnya. [[Pierre-Simon Laplace]] menyebut logaritma sebagai

 

 

Another critical application was the [[slide rule]], a pair of logarithmically divided scales used for calculation. The non-sliding logarithmic scale, [[Gunter's rule]], was invented shortly after Napier's invention. [[William Oughtred]] enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other. Numbers are placed on sliding scales at distances proportional to the differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms, as illustrated here:

[[Berkas:Slide_rule_example2_with_labels.svg|al=A slide rule: two rectangles with logarithmically ticked axes, arrangement to add the distance from 1 to 2 to the distance from 1 to 3, indicating the product 6.|pus|jmpl|550x550px|Schematic depiction of a slide rule. Starting from 2 on the lower scale, add the distance to 3 on the upper scale to reach the product&nbsp;6. The slide rule works because it is marked such that the distance from 1 to {{mvar|x}} is proportional to the logarithm of {{mvar|x}}.]]

 

 

A deeper study of logarithms requires the concept of a ''[[Function (mathematics)|function]]''. A function is a rule that, given one number, produces another number.<ref>{{citation|last1=Devlin|first1=Keith|author1-link=Keith Devlin|title=Sets, functions, and logic: an introduction to abstract mathematics|publisher=Chapman & Hall/CRC|location=Boca Raton, Fla|edition=3rd|series=Chapman & Hall/CRC mathematics|isbn=978-1-58488-449-1|year=2004|url={{google books |plainurl=y |id=uQHF7bcm4k4C}}}}, or see the references in [[Function (mathematics)|function]]</ref> An example is the function producing the {{mvar|x}}-th power of {{mvar|b}} from any real number&nbsp;{{mvar|x}}, where the base&nbsp;{{mvar|b}} is a fixed number. This function is written as {{math|1=''f''(''x'') = {{mvar|b}}^{&thinsp;''x''}}}. When {{mvar|b}} is positive and unequal to 1, we show below that {{Mvar|f}} is invertible when considered as a function from the reals to the positive reals.

 

 

It is a standard result in real analysis that any continuous strictly monotonic function is bijective between its domain and range. This fact follows from the [[intermediate value theorem]].<ref name="LangIII.3">{{Citation|last1=Lang|first1=Serge|title=Undergraduate analysis|year=1997|series=[[Undergraduate Texts in Mathematics]]|edition=2nd|location=Berlin, New York|publisher=[[Springer-Verlag]]|doi=10.1007/978-1-4757-2698-5|isbn=978-0-387-94841-6|mr=1476913|author1-link=Serge Lang}}, section III.3</ref> Now, {{mvar|f}} is [[Monotonic function|strictly increasing]] (for {{math|''b'' > 1}}), or strictly decreasing (for {{math|0 < {{mvar|b}} < 1}}),<ref name="LangIV.2">{{Harvard citations|last1=Lang|year=1997|nb=yes|loc=section IV.2}}</ref> is continuous, has domain <math>\R</math>, and has range <math>\R_{> 0}</math>. Therefore, {{Mvar|f}} is a bijection from <math>\R</math> to <math>\R_{>0}</math>. In other words, for each positive real number {{Mvar|y}}, there is exactly one real number {{Mvar|x}} such that <math>b^x = y</math>.

 

 

 

: <math>\log_b(xy) = \log_b x + \log_b y.</math>

 

 

: <math>f(xy)=f(x)+f(y).</math>

 

[[Berkas:Logarithm_inversefunctiontoexp.svg|al=The graphs of two functions.|ka|jmpl|The graph of the logarithm function {{math|log_''b''&thinsp;(''x'')}} (blue) is obtained by [[Reflection (mathematics)|reflecting]] the graph of the function {{math|''b''^''x''}} (red) at the diagonal line ({{math|1=''x'' = {{mvar|y}}}}).]]

As discussed above, the function {{math|log_''b''}} is the inverse to the exponential function <math>x\mapsto b^x</math>. Therefore, Their [[Graph of a function|graphs]] correspond to each other upon exchanging the {{mvar|x}}- and the {{mvar|y}}-coordinates (or upon reflection at the diagonal line {{Math|1=''x'' = ''y''}}), as shown at the right: a point {{math|1=(''t'', ''u'' = {{mvar|b}}^''t'')}} on the graph of {{Mvar|f}} yields a point {{math|1=(''u'', ''t'' = log_''b''&thinsp;''u'')}} on the graph of the logarithm and vice versa. As a consequence, {{math|log_''b''&thinsp;(''x'')}} [[Divergent sequence|diverges to infinity]] (gets bigger than any given number) if {{mvar|x}} grows to infinity, provided that {{mvar|b}} is greater than one. In that case, {{math|log_''b''(''x'')}} is an [[increasing function]]. For {{math|''b'' < 1}}, {{math|log_''b''&thinsp;(''x'')}} tends to minus infinity instead. When {{mvar|x}} approaches zero, {{math|log_''b''&thinsp;''x''}} goes to minus infinity for {{math|''b'' > 1}} (plus infinity for {{math|''b'' < 1}}, respectively).

 

[[Berkas:Logarithm_derivative.svg|al=A graph of the logarithm function and a line touching it in one point.|ka|jmpl|220x220px|The graph of the [[natural logarithm]] (green) and its tangent at {{math|''x'' {{=}} 1.5}} (black)]]

Analytic properties of functions pass to their inverses.<ref name="LangIII.3" /> Thus, as {{math|1=''f''(''x'') = {{mvar|b}}^''x''}} is a continuous and [[differentiable function]], so is {{math|log_''b''&thinsp;''y''}}. Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the [[derivative]] of {{math|''f''(''x'')}} evaluates to {{math|ln(''b'') ''b''^''x''}} by the properties of the [[exponential function]], the [[chain rule]] implies that the derivative of {{math|log_''b''&thinsp;''x''}} is given by<ref name="LangIV.2" /><ref>{{citation|work=Wolfram Alpha|title=Calculation of ''d/dx(Log(b,x))''|publisher=[[Wolfram Research]]|access-date=15 March 2011|url=http://www.wolframalpha.com/input/?i=d/dx(Log(b,x))}}</ref>

[[List of integrals of logarithmic functions|Related formulas]], such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.<ref>{{Harvard citations|editor1-last=Abramowitz|editor2-last=Stegun|year=1972|nb=yes|loc=p. 69}}</ref>

 

[[Berkas:Natural_logarithm_integral.svg|al=A hyperbola with part of the area underneath shaded in grey.|ka|jmpl|The [[natural logarithm]] of ''{{Mvar|t}}'' is the shaded area underneath the graph of the function {{math|1=''f''(''x'') = 1/''x''}} (reciprocal of {{mvar|x}}).]]

The [[natural logarithm]] of {{Mvar|t}} can be defined as the [[definite integral]]:

[[Limit of a sequence|converges]] (i.e. gets arbitrarily close) to a number known as the [[Euler–Mascheroni constant]] {{math|1=''γ'' = 0.5772...}}. This relation aids in analyzing the performance of algorithms such as [[quicksort]].<ref>{{Citation|last1=Havil|first1=Julian|title=Gamma: Exploring Euler's Constant|publisher=[[Princeton University Press]]|isbn=978-0-691-09983-5|year=2003}}, sections 11.5 and 13.8</ref>

 

[[Real number|Real numbers]] that are not [[Algebraic number|algebraic]] are called [[Transcendental number|transcendental]];<ref>{{citation|title=Selected papers on number theory and algebraic geometry|volume=172|first1=Katsumi|last1=Nomizu|author-link=Katsumi Nomizu|location=Providence, RI|publisher=AMS Bookstore|year=1996|isbn=978-0-8218-0445-2|page=21|url={{google books |plainurl=y |id=uDDxdu0lrWAC|page=21}}}}</ref> for example, [[Pi|{{pi}}]] and ''[[E (mathematical constant)|e]]'' are such numbers, but <math>\sqrt{2-\sqrt 3}</math> is not. [[Almost all]] real numbers are transcendental. The logarithm is an example of a [[transcendental function]]. The [[Gelfond–Schneider theorem]] asserts that logarithms usually take transcendental, i.e. "difficult" values.<ref>{{Citation|last1=Baker|first1=Alan|author1-link=Alan Baker (mathematician)|title=Transcendental number theory|publisher=[[Cambridge University Press]]|isbn=978-0-521-20461-3|year=1975}}, p.&nbsp;10</ref>

 

[[Berkas:Logarithm_keys.jpg|jmpl|The logarithm keys (LOG for base&nbsp;10 and LN for base&nbsp;{{mvar|e}}) on a [[TI-83 series|TI-83 Plus]] graphing calculator]]

Logarithms are easy to compute in some cases, such as {{math|1=log₁₀&thinsp;(1000) = 3}}. In general, logarithms can be calculated using [[power series]] or the [[arithmetic–geometric mean]], or be retrieved from a precalculated [[logarithm table]] that provides a fixed precision.<ref>{{Citation|last1=Muller|first1=Jean-Michel|title=Elementary functions|publisher=Birkhäuser Boston|location=Boston, MA|edition=2nd|isbn=978-0-8176-4372-0|year=2006}}, sections 4.2.2 (p. 72) and 5.5.2 (p. 95)</ref><ref>{{Citation|last1=Hart|last2=Cheney|last3=Lawson|year=1968|publisher=John Wiley|location=New York|title=Computer Approximations|series=SIAM Series in Applied Mathematics|display-authors=etal}}, section 6.3, pp.&nbsp;105–11</ref> [[Newton's method]], an iterative method to solve equations approximately, can also be used to calculate the logarithm, because its inverse function, the exponential function, can be computed efficiently.<ref>{{Citation|last1=Zhang|first1=M.|last2=Delgado-Frias|first2=J.G.|last3=Vassiliadis|first3=S.|title=Table driven Newton scheme for high precision logarithm generation|doi=10.1049/ip-cdt:19941268|journal=IEE Proceedings - Computers and Digital Techniques|issn=1350-2387|volume=141|year=1994|issue=5|pages=281–92}}, section 1 for an overview</ref> Using look-up tables, [[CORDIC]]-like methods can be used to compute logarithms by using only the operations of addition and [[Arithmetic shift|bit shifts]].<ref>{{Citation|url=https://semanticscholar.org/paper/b3741168ba25f23b694cf8f9c80fb4f2aabce513|first=J.E.|last=Meggitt|title=Pseudo Division and Pseudo Multiplication Processes|journal=IBM Journal of Research and Development|date=April 1962|doi=10.1147/rd.62.0210|volume=6|issue=2|pages=210–26|s2cid=19387286}}</ref><ref>{{Citation|last=Kahan|first=W.|author-link=William Kahan|title=Pseudo-Division Algorithms for Floating-Point Logarithms and Exponentials|date=20 May 2001}}</ref> Moreover, the [[Binary logarithm#Algorithm|binary logarithm algorithm]] calculates {{math|lb(''x'')}} [[Recursion|recursively]], based on repeated squarings of {{mvar|x}}, taking advantage of the relation

: <math>\log_2\left(x^2\right) = 2 \log_2 |x|.</math>

 

 

 

[[Berkas:Taylor_approximation_of_natural_logarithm.gif|al=An animation showing increasingly good approximations of the logarithm graph.|ka|jmpl|The Taylor series of {{math|ln(''z'')}} centered at {{math|''z'' {{=}} 1}}. The animation shows the first&nbsp;10 approximations along with the 99th and 100th. The approximations do not converge beyond a distance of 1 from the center.]]

If the logarithm of a large integer&nbsp;{{mvar|n}} is known, then this series yields a fast converging series for {{math|log(''n''+1)}}, with a [[rate of convergence]] of <math display="inline">\left(\frac{1}{2 n+1}\right)^{2}</math>.

 

The [[arithmetic–geometric mean]] yields high precision approximations of the [[natural logarithm]]. Sasaki and Kanada showed in 1982 that it was particularly fast for precisions between 400 and 1000 decimal places, while Taylor series methods were typically faster when less precision was needed. In their work {{math|ln(''x'')}} is approximated to a precision of {{math|2^−''p''}} (or {{Mvar|p}}&nbsp;precise bits) by the following formula (due to [[Carl Friedrich Gauss]]):<ref>{{Citation|first1=T.|last1=Sasaki|first2=Y.|last2=Kanada|title=Practically fast multiple-precision evaluation of log(x)|journal=Journal of Information Processing|volume=5|issue=4|pages=247–50|year=1982|url=http://ci.nii.ac.jp/naid/110002673332|access-date=30 March 2011}}</ref><ref>{{Citation|first1=Timm|title=Stacs 99|last1=Ahrendt|publisher=Springer|location=Berlin, New York|series=Lecture notes in computer science|doi=10.1007/3-540-49116-3_28|volume=1564|year=1999|pages=302–12|isbn=978-3-540-65691-3|chapter=Fast Computations of the Exponential Function}}</ref>

 

to ensure the required precision. A larger {{mvar|m}} makes the {{math|M(''x'', ''y'')}} calculation take more steps (the initial {{mvar|x}} and {{mvar|y}} are farther apart so it takes more steps to converge) but gives more precision. The constants {{math|{{pi}}}} and {{math|ln(2)}} can be calculated with quickly converging series.

 

While at [[Los Alamos National Laboratory]] working on the [[Manhattan Project]], [[Richard Feynman]] developed a bit-processing algorithm, to compute the logarithm, that is similar to long division and was later used in the [[Connection Machine]]. The algorithm uses the fact that every real number {{Math|1 < ''x'' < 2}} is representable as a product of distinct factors of the form {{Math|1 + 2^−''k''}}. The algorithm sequentially builds that product&nbsp;{{Mvar|P}}, starting with {{math|''P'' {{=}} 1}} and {{math|''k'' {{=}} 1}}: if {{math|''P'' · (1 + 2^−''k'') < ''x''}}, then it changes {{Mvar|P}} to {{math|''P'' · (1 + 2^−''k'')}}. It then increases <math>k</math> by one regardless. The algorithm stops when {{Mvar|k}} is large enough to give the desired accuracy. Because {{Math|log(''x'')}} is the sum of the terms of the form {{Math|log(1 + 2^−''k'')}} corresponding to those {{Mvar|k}} for which the factor {{Math|1 + 2^−''k''}} was included in the product&nbsp;{{Mvar|P}}, {{Math|log(''x'')}} may be computed by simple addition, using a table of {{Math|log(1 + 2^−''k'')}} for all {{Mvar|k}}. Any base may be used for the logarithm table.<ref>{{citation|first=Danny|last=Hillis|author-link=Danny Hillis|title=Richard Feynman and The Connection Machine|journal=Physics Today|volume=42|issue=2|page=78|date=15 January 1989|doi=10.1063/1.881196|bibcode=1989PhT....42b..78H}}</ref>

 

[[Berkas:NautilusCutawayLogarithmicSpiral.jpg|al=A photograph of a nautilus' shell.|jmpl|A [[nautilus]] displaying a logarithmic spiral]]

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|Logarithmic scales]] 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]].

 

{{Main|Logarithmic scale}}

[[Berkas:Germany_Hyperinflation.svg|al=A graph of the value of one mark over time. The line showing its value is increasing very quickly, even with logarithmic scale.|ka|jmpl|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]]]]

[[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|exponential functions]] 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|power laws]].<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>

 

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|series=Encyclopedia of Perception|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 distance to 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|url=https://semanticscholar.org/paper/3087289229146fc344560478aac366e4977749c0}}, 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|journal=Psychological Science|archive-date=17 May 2011|archive-url=https://web.archive.org/web/20110517002232/http://www.psy.cmu.edu/~siegler/sieglerbooth-cd04.pdf|access-date=7 January 2011|s2cid=9583202|citeseerx=10.1.1.727.3696|pmid=12741747|url=http://www.psy.cmu.edu/~siegler/sieglerbooth-cd04.pdf|year=2003|last1=Siegler|pages=237–43|issue=3|volume=14|title=The Development of Numerical Estimation. Evidence for Multiple Representations of Numerical Quantity|first2=John E.|last2=Opfer|first1=Robert S.|url-status=dead}}</ref><ref>{{Citation|last1=Dehaene|issue=5880|bibcode=2008Sci...320.1217D|journal=Science|year=2008|pmid=18511690|pmc=2610411|doi=10.1126/science.1156540|pages=1217–20|volume=320|first1=Stanislas|title=Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures|first4=Pierre|last4=Pica|first3=Elizabeth|last3=Spelke|first2=Véronique|last2=Izard|citeseerx=10.1.1.362.2390}}</ref>

 

[[Berkas:PDF-log_normal_distributions.svg|al=Three asymmetric PDF curves|ka|jmpl|Three [[Probability density function|probability density functions]] (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.]]

[[Berkas:Benfords_law_illustrated_by_world's_countries_population.png|al=A bar chart and a superimposed second chart. The two differ slightly, but both decrease in a similar fashion.|ka|jmpl|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.]]

[[Benford's law]] describes the occurrence of digits in many [[Data set|data sets]], 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|log₁₀&thinsp;(''d'' + 1) − log₁₀&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>

 

[[Analysis of algorithms]] is a branch of [[computer science]] that studies the [[Time complexity|performance]] of [[Algorithm|algorithms]] (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>

 

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|log₂&thinsp;''N'' + 1}}&nbsp;[[Bit|bits]]. In other words, the amount of [[Memory (computing)|memory]] needed to store {{mvar|N}} grows logarithmically with {{mvar|N}}.

 

[[Berkas:Chaotic_Bunimovich_stadium.png|al=An oval shape with the trajectories of two particles.|ka|jmpl|[[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.]]

[[Entropy]] is broadly a measure of the disorder of some system. In [[statistical thermodynamics]], the entropy&nbsp;''S'' of some physical system is defined as

[[Lyapunov exponent|Lyapunov exponents]] 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.

 

[[Berkas:Sierpinski_dimension.svg|al=Parts of a triangle are removed in an iterated way.|ka|jmpl|400x400px|The Sierpinski triangle (at the right) is constructed by repeatedly replacing [[Equilateral triangle|equilateral triangles]] by three smaller ones.]]

Logarithms occur in definitions of the [[Fractal dimension|dimension]] of [[Fractal|fractals]].<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|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.

 

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[[Natural logarithm|Natural logarithms]] 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|prime numbers]] 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

 

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>

 

 

{{Main|Complex logarithm}}

[[Berkas:Complex_number_illustration_multiple_arguments.svg|al=An illustration of the polar form: a point is described by an arrow or equivalently by its length and angle to the x-axis.|ka|jmpl|Polar form of {{math|''z {{=}} x + iy''}}. Both {{mvar|φ}} and {{mvar|φ'}} are arguments of {{mvar|z}}.]]

The illustration at the right depicts {{math|Log(''z'')}}, confining the arguments of {{mvar|z}} to the interval {{open-closed|−π, π}}. This way the corresponding branch of the complex logarithm has discontinuities all along the negative real {{mvar|x}}&nbsp;axis, which can be seen in the jump in the hue there. This discontinuity arises from jumping to the other boundary in the same branch, when crossing a boundary, i.e. not changing to the corresponding {{mvar|k}}-value of the continuously neighboring branch. Such a locus is called a [[branch cut]]. Dropping the range restrictions on the argument makes the relations "argument of {{mvar|z}}", and consequently the "logarithm of {{mvar|z}}", [[Multi-valued function|multi-valued functions]].

 

Exponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm. For example, the [[logarithm of a matrix]] is the (multi-valued) inverse function of the [[matrix exponential]].<ref>{{Citation|last1=Higham|first1=Nicholas|author1-link=Nicholas Higham|title=Functions of Matrices. Theory and Computation|location=Philadelphia, PA|publisher=[[Society for Industrial and Applied Mathematics|SIAM]]|isbn=978-0-89871-646-7|year=2008}}, chapter 11.</ref> Another example is the [[P-adic logarithm function|''p''-adic logarithm]], the inverse function of the [[P-adic exponential function|''p''-adic exponential]]. Both are defined via Taylor series analogous to the real case.<ref>{{Neukirch ANT|mode=cs2}}, section II.5.</ref> In the context of [[differential geometry]], the [[Exponential map (Riemannian geometry)|exponential map]] maps the [[tangent space]] at a point of a [[Differentiable manifold|manifold]] to a [[Neighborhood (mathematics)|neighborhood]] of that point. Its inverse is also called the logarithmic (or log) map.<ref>{{Citation|last1=Hancock|first1=Edwin R.|last2=Martin|first2=Ralph R.|last3=Sabin|first3=Malcolm A.|title=Mathematics of Surfaces XIII: 13th IMA International Conference York, UK, September 7–9, 2009 Proceedings|url=https://books.google.com/books?id=0cqCy9x7V_QC&pg=PA379|publisher=Springer|year=2009|page=379|isbn=978-3-642-03595-1}}</ref>

 

{{anchor|double logarithm}}Further logarithm-like inverse functions include the ''double logarithm''&nbsp;{{math|ln(ln(''x''))}}, the ''[[Super-logarithm|super- or hyper-4-logarithm]]'' (a slight variation of which is called [[iterated logarithm]] in computer science), the [[Lambert W function]], and the [[logit]]. They are the inverse functions of the [[double exponential function]], [[tetration]], of {{math|''f''(''w'') {{=}} ''we^w''}},<ref>{{Citation|last1=Corless|year=1996|archive-date=14 December 2010|archive-url=https://web.archive.org/web/20101214110615/http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdf|access-date=13 February 2011|s2cid=29028411|doi=10.1007/BF02124750|pages=329–59|volume=5|issn=1019-7168|journal=Advances in Computational Mathematics|url=http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdf|first1=R.|title=On the Lambert ''W'' function|author5-link=Donald Knuth|first5=Donald|last5=Knuth|first4=D.|last4=Jeffrey|first3=D.|last3=Hare|first2=G.|last2=Gonnet|url-status=dead}}</ref> and of the [[logistic function]], respectively.<ref>{{Citation|last1=Cherkassky|first1=Vladimir|last2=Cherkassky|first2=Vladimir S.|last3=Mulier|first3=Filip|title=Learning from data: concepts, theory, and methods|publisher=[[John Wiley & Sons]]|location=New York|series=Wiley series on adaptive and learning systems for signal processing, communications, and control|isbn=978-0-471-68182-3|year=2007}}, p.&nbsp;357</ref>

 

From the perspective of [[group theory]], the identity {{math|log(''cd'') {{=}} log(''c'') + log(''d'')}} expresses a [[group isomorphism]] between positive [[Real number|reals]] under multiplication and reals under addition. Logarithmic functions are the only continuous isomorphisms between these groups.<ref>{{Citation|last1=Bourbaki|first1=Nicolas|author1-link=Nicolas Bourbaki|title=General topology. Chapters 5–10|publisher=[[Springer-Verlag]]|location=Berlin, New York|series=Elements of Mathematics|isbn=978-3-540-64563-4|mr=1726872|year=1998}}, section V.4.1</ref> By means of that isomorphism, the [[Haar measure]] ([[Lebesgue measure]])&nbsp;{{math|''dx''}} on the reals corresponds to the Haar measure&nbsp;{{math|''dx''/''x''}} on the positive reals.<ref>{{Citation|last1=Ambartzumian|first1=R.V.|author-link=Rouben V. Ambartzumian|title=Factorization calculus and geometric probability|publisher=[[Cambridge University Press]]|isbn=978-0-521-34535-4|year=1990|url-access=registration|url=https://archive.org/details/factorizationcal0000amba}}, section 1.4</ref> The non-negative reals not only have a multiplication, but also have addition, and form a [[semiring]], called the [[probability semiring]]; this is in fact a [[semifield]]. The logarithm then takes multiplication to addition (log multiplication), and takes addition to log addition ([[LogSumExp]]), giving an [[isomorphism]] of semirings between the probability semiring and the [[log semiring]].

 

It is related to the [[natural logarithm]] by {{math|1=Li₁&thinsp;(''z'') = −ln(1 − ''z'')}}. Moreover, {{math|Li_''s''&thinsp;(1)}} equals the [[Riemann zeta function]] {{math|ζ(''s'')}}.<ref>{{dlmf|id=25.12|first=T.M.|last=Apostol}}</ref>

 

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* [[Logarithmic notation]]

 

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