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Logarithm
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==Particular bases<span class="anchor" id="log_base_anchor"></span>== [[File:Log4.svg|thumb|upright=1.2|Overlaid graphs of the logarithm for bases {{sfrac| 1 | 2 }}, 2, and {{mvar|e}}]] Among all choices for the base, three are particularly common. These are {{math| ''b'' {{=}} 10}}, {{math| ''b'' {{=}} [[e (mathematical constant)|''e'']]}} (the [[Irrational number|irrational]] mathematical constant {{nobr|{{math|[[e (mathematical constant)|''e'']] ≈ 2.71828183 }} ),}} and {{math| ''b'' {{=}} 2}} (the [[binary logarithm]]). In [[mathematical analysis]], the logarithm base {{mvar|e}} is widespread because of analytical properties explained below. On the other hand, {{nobr|base 10}} logarithms (the [[common logarithm]]) are easy to use for manual calculations in the [[decimal]] number system:<ref> {{cite book |last=Downing |first=Douglas |year=2003 |title=Algebra the Easy Way |at=chapter 17, p. 275 |series=Barron's Educational Series |location=Hauppauge, NY |publisher=Barron's |isbn=978-0-7641-1972-9 |url=https://archive.org/details/algebraeasyway00down_0 }} </ref> <math display=block>\log_{10}\,(\,10\,x\,)\ =\;\log_{10} 10\ +\;\log_{10} x\ =\ 1\,+\,\log_{10} x\,.</math> Thus, {{math|log<sub>10</sub> (''x'')}} is related to the number of [[decimal digit]]s of a positive integer {{mvar|x}}: The number of digits is the smallest [[integer]] strictly bigger than {{nobr| {{math| log<sub>10</sub> (''x'')}} .}}<ref> {{cite book |last = Wegener |first = Ingo |year = 2005 |title = Complexity Theory: Exploring the limits of efficient algorithms |publisher = [[Springer-Verlag]] |location = Berlin, DE / New York, NY |isbn = 978-3-540-21045-0 |page = 20 }} </ref> For example, {{math|log<sub>10</sub>(5986)}} is approximately 3.78 . The [[Floor and ceiling functions|next integer above]] it is 4, which is the number of digits of 5986. Both the natural logarithm and the binary logarithm are used in [[information theory]], corresponding to the use of [[nat (unit)|nat]]s or [[bit]]s as the fundamental units of information, respectively.<ref> {{cite book |first = Jan C.A. |last = van der Lubbe |year = 1997 |title = Information Theory |publisher = Cambridge University Press |isbn = 978-0-521-46760-5 |page = 3 |url = {{google books |plainurl=y |id=tBuI_6MQTcwC|page=3}} }} </ref> Binary logarithms are also used in [[computer science]], where the [[binary numeral system|binary system]] is ubiquitous; in [[music theory]], where a pitch ratio of two (the [[octave]]) is ubiquitous and the number of [[cent (music)|cents]] between any two pitches is a scaled version of the binary logarithm, or log 2 times 1200, of the pitch ratio (that is, 100 cents per [[semitone]] in [[12 tone equal temperament|conventional equal temperament]]), or equivalently the log base {{nobr| {{math|2{{sup|1/1200}} }} ;}} and in [[photography]] rescaled base 2 logarithms are used to measure [[exposure value]]s, [[luminance|light levels]], [[exposure time]]s, lens [[aperture]]s, and [[film speed]]s in "stops".<ref> {{cite book |first1 = Elizabeth |last1 = Allen |first2 = Sophie |last2 = Triantaphillidou |year = 2011 |title = The Manual of Photography |publisher = Taylor & Francis |isbn = 978-0-240-52037-7 |page = 228 |url = {{google books |plainurl=y |id=IfWivY3mIgAC|page=228}} }} </ref> The abbreviation {{math|log ''x''}} is often used when the intended base can be inferred based on the context or discipline, or when the base is indeterminate or immaterial. Common logarithms (base 10), historically used in logarithm tables and slide rules, are a basic tool for measurement and computation in many areas of science and engineering; in these contexts {{math|log ''x''}} still often means the base ten logarithm.<ref> {{cite book |first = David F. |last = Parkhurst |year = 2007 |title = Introduction to Applied Mathematics for Environmental Science |edition=illustrated |publisher = Springer Science & Business Media |isbn = 978-0-387-34228-3 |page = 288 |url = {{google books |plainurl=y |id=h6yq_lOr8Z4C|page=288 }} }} </ref> In mathematics {{math|log ''x''}} usually refers to the natural logarithm (base {{mvar|e}}).<ref>{{cite book |last = Rudin |first = Walter |year = 1984 |section = Theorem 3.29 |title = Principles of Mathematical Analysis |edition=3rd ed., International student |publisher = McGraw-Hill International |location = Auckland, NZ |isbn = 978-0-07-085613-4 |url = https://archive.org/details/principlesofmath00rudi }} </ref> In computer science and information theory, {{math|log}} often refers to binary logarithms (base 2).<ref> {{cite book |first1 = Michael T. |last1 = Goodrich |author1-link = Michael T. Goodrich |first2 = Roberto |last2 = Tamassia |author2-link = Roberto Tamassia |year = 2002 |title = Algorithm Design: Foundations, analysis, and internet examples |publisher = John Wiley & Sons |page = 23 |quote = One of the interesting and sometimes even surprising aspects of the analysis of data structures and algorithms is the ubiquitous presence of logarithms ... As is the custom in the computing literature, we omit writing the base {{mvar|b}} of the logarithm when {{nobr| {{math| ''b'' {{=}} 2}} .}} }} </ref> The following table lists common notations for logarithms to these bases. The "ISO notation" column lists designations suggested by the [[International Organization for Standardization]].<ref> {{cite report |department = Quantities and units |title = {{grey|[title not cited]}} |section = Part 2: Mathematics |year = 2019 |id = {{nobr| [[ISO 80000-2]]:2019 }} / {{nobr| EN [[ISO 80000-2]] }} |publisher=[[International Organization for Standardization]] }} : ''See also'' [[ISO 80000-2]] . </ref> {| class="wikitable" style="text-align:center;" |- !scope="col"| Base {{mvar|b}} !scope="col"| Name for log<sub>''b''</sub> ''x'' !scope="col"| ISO notation !scope="col"| Other notations |- !scope="row"| 2 | [[binary logarithm]] | {{math|lb ''x''}} <ref name=gullberg> {{cite book | last = Gullberg | first = Jan | year = 1997 | title = Mathematics: From the birth of numbers | location = New York, NY | publisher = W.W. Norton & Co | isbn = 978-0-393-04002-9 | url-access = registration | url = https://archive.org/details/mathematicsfromb1997gull }} </ref> | {{math|ld ''x''}}, {{math|log ''x''}}, {{math|lg ''x''}},<ref>{{citation | title=The Chicago Manual of Style | year=2003 | edition=25th | publisher=University of Chicago Press | page=530 | title-link=The Chicago Manual of Style }}.</ref> {{math|log<sub>2</sub> ''x''}} |- ! scope="row"|{{mvar|e}} | [[natural logarithm]] | {{math|ln ''x''}} {{refn|name=adaa|group=nb|z Some mathematicians disapprove of this notation. In his 1985 autobiography, [[Paul Halmos]] criticized what he considered the "childish {{math|ln}} notation", which he said no mathematician had ever used.<ref> {{cite book |first = P. |last = Halmos |author-link = Paul Halmos |year = 1985 |title = I Want to be a Mathematician: An automathography |publisher = Springer-Verlag |location = Berlin, DE / New York, NY |isbn = 978-0-387-96078-4 }} </ref> The notation was invented by the 19th century mathematician [[Irving Stringham|I. Stringham]].<ref> {{cite book |first = I. |last = Stringham |author-link=Irving Stringham |year = 1893 |title = Uniplanar Algebra |publisher = The Berkeley Press |page = {{mvar|xiii}} |url = {{google books |plainurl=y |id=hPEKAQAAIAAJ|page=13}} |quote = Being part I of a propædeutic to the higher mathematical analysis }} </ref><ref> {{cite book |first = Roy S. |last=Freedman |year = 2006 |title = Introduction to Financial Technology |publisher = Academic Press |location=Amsterdam |isbn=978-0-12-370478-8 |page = 59 |url = {{google books |plainurl=y |id=APJ7QeR_XPkC|page=5}} }} </ref> }} | {{math|log {{mvar|x}}}}, {{math|log<sub>''e''</sub> ''x''}} |- !scope="row"| 10 | [[common logarithm]] | {{math|lg ''x''}} | {{math|log ''x''}}, {{math|log<sub>10</sub> ''x''}} |- !scope="row"| {{mvar|b}} | logarithm to base {{mvar|b}} | {{math|log<sub>''b''</sub> ''x''}} | |}
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