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| {{nofootnotes|date=May 2012}}
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| [[File:Graph of common logarithm.svg|right|thumb|400px|The common logarithm.]] | |
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| In [[mathematics]], the '''common logarithm''' is the [[logarithm]] with base 10. It is also known as the '''decadic logarithm''' and also as the '''decimal logarithm''', named after its base, or '''Briggsian logarithm''', after [[Henry Briggs (mathematician)|Henry Briggs]], an English mathematician who pioneered its use. It is indicated by log<sub>10</sub>(''x''), or sometimes Log(''x'') with a capital ''L'' (however, this notation is ambiguous since it can also mean the complex natural logarithmic [[multi-valued function]]). On calculators it is usually "log", but [[mathematician]]s usually mean [[natural logarithm]] rather than common logarithm when they write "log". To mitigate this ambiguity the [[ISO 31-11|ISO specification]] is that log<sub>10</sub>(''x'') should be lg (''x'') and log<sub>''e''</sub>(''x'') should be ln (''x'').
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| ==Uses==
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| Before the early 1970s, handheld electronic [[calculator]]s were not yet in widespread use. Due to their utility in saving work in laborious multiplications and divisions with pen and paper, [[mathematical table|tables]] of base 10 [[logarithm]]s were given in appendices of many books. Such a table of "common logarithms" gave the logarithm, often to 4 or 5 decimal places, of each number in the left-hand column, which ran from 1 to 10 by small increments, perhaps 0.01 or 0.001. There was only a need to include numbers between 1 and 10, since the logarithms of larger numbers were easily calculated.
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| For example, the logarithm of 120 is given by:
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| :<math>\log_{10}120=\log_{10}(10^2\times 1.2)=2+\log_{10}1.2\approx2+0.079181.</math>
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| The last number (0.079181)—the fractional part of the logarithm of 120, known as the '''mantissa''' of the common logarithm of 120—was found in the table.<ref group="note">This stems from an older, non-numerical, meaning of the word ''mantissa'': a minor addition or supplement, e.g. to a text. For a more modern use of the word ''mantissa'', see [[significand]].</ref> The location of the decimal point in 120 tells us that the integer part of the common logarithm of 120, called the '''characteristic''' of the common logarithm of 120, is 2.
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| Numbers between (and excluding) 0 and 1 have negative logarithms. For example,
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| :<math>\log_{10}0.012=\log_{10}(10^{-2}\times 1.2)=-2+\log_{10}1.2\approx-2+0.079181=-1.920819</math>
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| To avoid the need for separate tables to convert positive and negative logarithms back to their original numbers, a bar notation is used:
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| :<math>\log_{10}0.012\approx-2+0.079181=\bar{2}.079181</math>
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| The bar over the characteristic indicates that it is negative whilst the mantissa remains positive.
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| {| class="wikitable" style="text-align:center;" border="1" cellpadding=5px
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| |+Common logarithm, characteristic, and mantissa of powers of 10 times a number
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| !number
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| !logarithm
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| !characteristic
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| !mantissa
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| !combined form
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| |-
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| !''n'' (= 5 × 10<sup>''i''</sup>)
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| !log<sub>10</sub>(''n'')
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| !''i'' (= floor(log<sub>10</sub>(''n'')) )
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| !log<sub>10</sub>(''n'') − characteristic
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| !
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| |-
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| |5 000 000
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| |6.698 970...
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| |6
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| |0.698 970...
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| |6.698 970...
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| |-
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| |50
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| |1.698 970...
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| |1
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| |0.698 970...
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| |1.698 970...
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| |-
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| |5
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| |0.698 970...
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| |0
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| |0.698 970...
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| |0.698 970...
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| |-
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| |0.5
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| |−0.301 029...
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| |−1
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| |0.698 970...
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| |{{overline|1}}.698 970...
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| |-
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| |0.000 005
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| |−5.301 029...
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| |−6
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| |0.698 970...
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| |{{overline|6}}.698 970...
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| |-
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| |}
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| Note that the mantissa is common to all of the 5×10<sup>''i''</sup>.
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| A [[Logarithm#Tables_of_logarithms|table of logarithms]] will have a single indexed entry for the same mantissa.
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| In the example, 0.698 970 (004 336 018 ...) will be listed once indexed by 5, or perhaps by 0.5 or by 500 etc..
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| The following example uses the bar notation to calculate 0.012 × 0.85 = 0.0102:
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| :<math>\begin{array}{rll}
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| \text{As found above,} &\log_{10}0.012\approx\bar{2}.079181 \\
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| \text{Since}\;\;\log_{10}0.85&=\log_{10}(10^{-1}\times 8.5)=-1+\log_{10}8.5&\approx-1+0.929419=\bar{1}.929419\;, \\
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| \log_{10}(0.012\times 0.85) &=\log_{10}0.012+\log_{10}0.85 &\approx\bar{2}.079181+\bar{1}.929419 \\
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| &=(-2+0.079181)+(-1+0.929419) &=-(2+1)+(0.079181+0.929419) \\
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| &=-3+1.008600 &=-2+0.008600\;^* \\
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| &\approx\log_{10}(10^{-2})+\log_{10}(1.02) &=\log_{10}(0.01\times 1.02) \\
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| &=\log_{10}(0.0102)
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| \end{array}</math>
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| <nowiki>*</nowiki> This step makes the mantissa between 0 and 1, so that its antilog (10<sup>mantissa</sup>) can be looked up.
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| [[Image:Slide rule example2.svg|thumb|500px|Numbers are placed on [[slide rule]] scales at distances proportional to the differences between their logarithms. By mechanically adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale, one can quickly determine that 2 x 3 = 6.]]
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| == History ==
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| Common logarithms are sometimes also called "Briggsian logarithms" after [[Henry Briggs (mathematician)|Henry Briggs]], a 17th-century British mathematician.
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| Because base 10 logarithms were most useful for computations, engineers generally simply wrote "log(''x'')" when they meant log<sub>10</sub>(''x''). Mathematicians, on the other hand, wrote "log(''x'')" when they meant log<sub>''e''</sub>(''x'') for the natural logarithm. Today, both notations are found. Since hand-held electronic calculators are designed by engineers rather than mathematicians, it became customary that they follow engineers' notation. So the notation, according to which one writes "ln(''x'')" when the natural logarithm is intended, may have been further popularized by the very invention that made the use of "common logarithms" far less common, electronic calculators.
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| == Numeric value ==
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| The numerical value for logarithm to the base 10 can be calculated with the following identity.
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| : <math> \log_{10}(x) = \frac{\ln(x)}{\ln(10)} \qquad \text{ or } \qquad \log_{10}(x) = \frac{\log_2(x)}{\log_2(10)}</math>
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| as procedures exist for determining the numerical value for [[natural logarithm|logarithm base ''e'']] and [[binary logarithm|logarithm base 2]].
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| * [[Natural logarithm#Numerical value]]
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| * [[Binary logarithm#Algorithm|Algorithms for computing binary logarithms]]
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| ==See also==
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| *[[logarithm#History|History of logarithms]]
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| ==Notes==
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| {{reflist|group="note"}}
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| {{reflist}}
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| ==References==
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| *Michael Möser: ''Engineering Acoustics: An Introduction to Noise Control''. Springer 2009, ISBN 978-3-540-92722-8, p. 448 ({{Google books|kl2TA53Ysh4C|restricted online copy|page=448}})
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| *A. D. Poliyanin, A. V. Manzhirov: ''Handbook of mathematics for engineers and scientists''. CRC Press 2007, ISBN 978-1-58488-502-3, p. 9 ({{Google books|ge6nk9W0BCcC|restricted online copy|page=9}})
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| ==External links==
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| *{{planetmath reference|id=8865|title=Briggsian logarithms}} includes a detailed example of using logarithm tables
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| [[Category:Logarithms]]
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