Editing Common logarithm

{{Short description|Mathematical function}}
{{More citations needed|date=August 2020}}[[File:Graph of common logarithm.svg|right|thumb|alt=The graph shows that log base ten of x rapidly approaches minus infinity as x approaches zero, but gradually rises to the value two as x approaches one hundred.|A graph of the common logarithm of numbers from 0.1 to 100]]

In [[mathematics]], the '''common logarithm''' (aka "standard logarithm") is the [[logarithm]] with base 10.<ref name="Hall_1909"/> It is also known as the '''decadic logarithm''', the '''decimal logarithm''' and the '''Briggsian logarithm'''. The name "Briggsian logarithm" is in honor of the British mathematician [[Henry Briggs (mathematician)|Henry Briggs]] who conceived of and developed the values for the "common logarithm". Historically', the "common logarithm" was known by its Latin name ''logarithmus decimalis''<ref name="Euler_1748"/> or ''logarithmus decadis''.<ref name="Scherffer_1772"/>

The mathematical notation for using the common logarithm is {{math|log(''x'')}},<ref>{{Cite web|title=Introduction to Logarithms|url=https://www.mathsisfun.com/algebra/logarithms.html|access-date=2020-08-29|website=www.mathsisfun.com}}</ref> {{math|log<sub>10</sub>(''x'')}},<ref name=":0">{{Cite web|last=Weisstein|first=Eric W.|title=Common Logarithm|url=https://mathworld.wolfram.com/CommonLogarithm.html|access-date=2020-08-29|website=mathworld.wolfram.com|language=en}}</ref> or sometimes {{math|Log(''x'')}} with a capital {{Math|L}};{{refn|group=lower-alpha|The notation {{math|Log}} is ambiguous, as this can also mean the complex natural logarithmic [[multi-valued function]].}} on [[calculator]]s, it is printed as "log", but mathematicians usually mean [[natural logarithm]] (logarithm with base {{mvar|e}} ≈ 2.71828) rather than common logarithm when writing "log".

[[File:APN2002 Table 1, 1000-1500.agr.tiff|thumb|300px|Page from a table of common logarithms. This page shows the logarithms for numbers from 1000 to 1509 to five decimal places. The complete table covers values up to 9999.]]
Before the early 1970s, handheld electronic calculators were not available, and [[mechanical calculator]]s capable of multiplication were bulky, expensive and not widely available. Instead, [[mathematical table|tables]] of base-10 logarithms were used in science, engineering and navigation—when calculations required greater accuracy than could be achieved with a [[slide rule]]. By turning multiplication and division to addition and subtraction, use of logarithms avoided laborious and error-prone paper-and-pencil multiplications and divisions.<ref name="Hall_1909"/> Because logarithms were so useful, [[mathematical table|table]]s of base-10 logarithms were given in appendices of many textbooks. Mathematical and navigation handbooks included tables of the logarithms of [[trigonometric function]]s as well.<ref name="Hedrick_1913"/> For the history of such tables, see [[log table]].

==Mantissa and characteristic==
An important property of base-10 logarithms, which makes them so useful in calculations, is that the logarithm of numbers greater than 1 that differ by a factor of a power of 10 all have the same [[fractional part]]. The fractional part is known as the '''mantissa'''.{{refn|group=lower-alpha|This use of the word ''mantissa'' stems from an older, non-numerical, meaning: a minor addition or supplement, e.g., to a text.{{citation needed|date=November 2024}} The word was introduced by [[Henry Briggs (mathematician)|Henry Briggs]].<ref>{{Cite book |last=Schwartzman |first=Steven |url=https://books.google.com/books?id=PsH2DwAAQBAJ |title=The Words of Mathematics: An Etymological Dictionary of Mathematical Terms in English |date=1994-12-31 |publisher=American Mathematical Soc. |isbn=978-1-61444-501-2 |pages=131 |language=en}}</ref> The word "mantissa" is often used to describe the part of a [[floating-point]] number that represents its [[Significant figures|significant digits]], although "[[significand]]" was the term used for this by [[IEEE 754]], and may be preferred to avoid confusion with logarithm mantissas.}} Thus, log tables need only show the fractional part. Tables of common logarithms typically listed the mantissa, to four or five decimal places or more, of each number in a range, e.g. 1000 to 9999.

The integer part, called the '''characteristic''', can be computed by simply counting how many places the decimal point must be moved, so that it is just to the right of the first significant digit. For example, the logarithm of 120 is given by the following calculation:

:<math>\log_{10}(120) = \log_{10}\left(10^2 \times 1.2\right) = 2 + \log_{10}(1.2) \approx 2 + 0.07918.</math>

The last number (0.07918)—the fractional part or the mantissa of the common logarithm of 120—can be found in the table shown. The location of the decimal point in 120 tells us that the integer part of the common logarithm of 120, the characteristic, is&nbsp;2.

===Negative logarithms===
Positive numbers less than 1 have negative logarithms. For example,

:<math>\log_{10}(0.012) = \log_{10}\left(10^{-2} \times 1.2\right) = -2 + \log_{10}(1.2) \approx -2 + 0.07918 = -1.92082.</math>

To avoid the need for separate tables to convert positive and negative logarithms back to their original numbers, one can express a negative logarithm as a negative integer characteristic plus a positive mantissa. To facilitate this, a special notation, called ''bar notation,'' is used:

:<math>\log_{10}(0.012) \approx \bar{2} + 0.07918 = -1.92082.</math>

The bar over the characteristic indicates that it is negative, while the mantissa remains positive. When reading a number in bar notation out loud, the symbol <math>\bar{n}</math> is read as "bar {{Mvar|n}}", so that <math>\bar{2}.07918</math> is read as "bar 2 point 07918...".  An alternative convention is to express the logarithm modulo 10, in which case

:<math>\log_{10}(0.012) \approx 8.07918 \bmod 10,</math>

with the actual value of the result of a calculation determined by knowledge of the reasonable range of the result.{{refn|group=lower-alpha|For example, {{cite journal
|year = 1825
|last = Bessel |first = F. W. |authorlink = Friedrich Bessel
|title = Über die Berechnung der geographischen Längen und Breiten aus geodätischen Vermessungen
|journal = Astronomische Nachrichten
|volume = 331 |issue = 8 |pages = 852–861
|doi = 10.1002/asna.18260041601
|arxiv = 0908.1823
|bibcode = 1825AN......4..241B
|s2cid = 118630614 }}
gives (beginning of section 8) <math>\log b = 6.51335464</math>,
<math>\log e = 8.9054355</math>.
From the context, it is understood that
<math> b = 10^{6.51335464}</math>, the minor radius of the earth ellipsoid
in [[toise]] (a large number), whereas
<math> e = 10^{8.9054355-10}</math>, the eccentricity of the earth ellipsoid
(a small number).}}

The following example uses the bar notation to calculate 0.012 &times; 0.85 = 0.0102:

:<math>\begin{array}{rll}
 \text{As found above,}          &  \log_{10}(0.012) \approx\bar{2}.07918\\
 \text{Since}\;\;\log_{10}(0.85) &= \log_{10}\left(10^{-1}\times 8.5\right) = -1 + \log_{10}(8.5) &\approx -1 + 0.92942 = \bar{1}.92942\\
 \log_{10}(0.012 \times 0.85)    &= \log_{10}(0.012) + \log_{10}(0.85)                            &\approx \bar{2}.07918 + \bar{1}.92942\\
                                 &= (-2 + 0.07918) + (-1 + 0.92942)                               &= -(2 + 1) + (0.07918 + 0.92942)\\
                                 &= -3 + 1.00860                                                  &= -2 + 0.00860\;^*\\
                                 &\approx \log_{10}\left(10^{-2}\right) + \log_{10}(1.02)         &= \log_{10}(0.01 \times 1.02)\\
                                 &= \log_{10}(0.0102).
\end{array}</math>

<nowiki>*</nowiki> This step makes the mantissa between 0 and 1, so that its [[antilog]] (10{{sup|mantissa}}) can be looked up.

The following table shows how the same mantissa can be used for a range of numbers differing by powers of ten:
{| class="wikitable" style="text-align:center;" border="1" cellpadding=5px
|+ Common logarithm, characteristic, and mantissa of powers of 10 times a number
! Number
! Logarithm
! Characteristic
! Mantissa
! Combined form
|-
! ''n'' = 5 × 10{{sup|''i''}}
! log{{sub|10}}(''n'')
! ''i'' = floor(log{{sub|10}}(''n''))
! log{{sub|10}}(''n'') − ''i''
!
|-
| 5 000 000
| 6.698 970...
| 6
| 0.698 970...
| 6.698 970...
|-
| 50
| 1.698 970...
| 1
| 0.698 970...
| 1.698 970...
|-
| 5
| 0.698 970...
| 0
| 0.698 970...
| 0.698 970...
|-
| 0.5
| −0.301 029...
| −1
| 0.698 970...
| {{overline|1}}.698 970...
|-
| 0.000 005
| −5.301 029...
| −6
| 0.698 970...
| {{overline|6}}.698 970...
|}

Note that the mantissa is common to all of the {{Math|5 {{times}} 10<sup>''i''</sup>}}. This holds for any positive [[real number|real number&nbsp;]]<math>x</math> because
:<math>\log_{10}\left(x \times10^i\right) = \log_{10}(x) + \log_{10}\left(10^i\right) = \log_{10}(x) + i.</math>

Since {{Mvar|i}} is a constant, the mantissa comes from <math>\log_{10}(x)</math>, which is constant for given <math>x</math>. This allows a [[Log table|table of logarithms]] to include only one entry for each mantissa. In the example of {{Math|5 {{times}} 10<sup>''i''</sup>}}, 0.698&nbsp;970 (004&nbsp;336&nbsp;018&nbsp;...) will be listed once indexed by 5 (or 0.5, or 500, etc.).

[[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 {{Math|1=2 {{times}} 3 = 6}}.]]

== History ==
{{Main|History of logarithms}}
Common logarithms are sometimes also called "Briggsian logarithms" after [[Henry Briggs (mathematician)|Henry Briggs]], a 17th&nbsp;century British mathematician. In 1616 and 1617, Briggs visited [[John Napier]] at [[Edinburgh]], the inventor of what are now called natural (base-''e'') logarithms, in order to suggest a change to Napier's logarithms. During these conferences, the alteration proposed by Briggs was agreed upon; and after his return from his second visit, he published the first [[chiliad]] of his logarithms.

Because base-10 logarithms were most useful for computations, engineers generally simply wrote "{{math|log(''x'')}}" when they meant {{math|log<sub>10</sub>(''x'')}}. Mathematicians, on the other hand, wrote "{{math|log(''x'')}}" when they meant {{math|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 "{{math|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.

To mitigate the ambiguity, the [[ISO 80000-2|ISO 80000 specification]] recommends that {{math|log<sub>''e''</sub>(''x'')}} should be {{math|ln(''x'')}}, while {{math|log<sub>10</sub>(''x'')}} should be written {{math|lg(''x'')}}, which unfortunately is used for the [[base-2 logarithm]] by CLRS and Sedgwick and ''[[The Chicago Manual of Style]]''.<ref name="clrs">{{citation 
| last1 = Cormen | first1 = Thomas H. | author1-link = Thomas H. Cormen
| last2 = Leiserson | first2 = Charles E. | author2-link = Charles E. Leiserson
| last3 = Rivest | first3 = Ronald L. | author3-link = Ron Rivest
| last4 = Stein | first4 = Clifford | author4-link = Clifford Stein
| title = Introduction to Algorithms
| orig-year = 1990
| year = 2001
| edition = 2nd
| publisher = MIT Press and McGraw-Hill
| isbn = 0-262-03293-7
| pages = 34, 53–54
| title-link = Introduction to Algorithms }}</ref><ref name="sw11">{{citation
 | title=Algorithms
 | first1=Robert | last1=Sedgewick | author1-link=Robert Sedgewick (computer scientist)
 | first2=Kevin Daniel | last2=Wayne
 | publisher=Addison-Wesley Professional
 | year=2011
 | isbn=978-0-321-57351-3
 | page=185
 |url = https://books.google.com/books?id=MTpsAQAAQBAJ&pg=PA185
}}.</ref><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>  <!-- I'd like to comment here that it's very rare for anyone to ''actually'' use "lg" to mean "<math>\log_{10}</math>", but I don't know where I would even start looking for sources for that assertion. -->

== Numeric value ==
[[File:Logarithm keys.jpg|thumb|The logarithm keys (''log'' for base-10 and {{math|ln}} for base-{{mvar|e}}) on a typical scientific calculator. The advent of hand-held calculators largely eliminated the use of common logarithms as an aid to computation.]]
The numerical value for logarithm to the base&nbsp;10 can be calculated with the following identities:<ref name=":0"/>

: <math> \log_{10}(x) = \frac{\ln(x)}{\ln(10)} \quad</math>  or  <math>\quad \log_{10}(x) = \frac{\log_2(x)}{\log_2(10)} \quad</math>  or  <math>\quad \log_{10}(x) = \frac{\log_B(x)}{\log_B(10)} \quad</math>

using logarithms of any available base <math>\, B ~.</math>

as procedures exist for determining the numerical value for [[natural logarithm|logarithm base {{mvar|e}}]] (see {{Section link|Natural logarithm|Efficient computation}}) and [[binary logarithm|logarithm base 2]] (see [[Binary logarithm#Algorithm|Algorithms for computing binary logarithms]]).

== Derivative ==
The derivative of a logarithm with a base ''b'' is such that<ref>{{Cite web|date=2021-04-14|title=Derivatives of Logarithmic Functions|url=https://www.math24.net/derivatives-logarithmic-functions|url-status=live|website=Math24|archive-url=https://web.archive.org/web/20201001225717/https://www.math24.net/derivatives-logarithmic-functions/ |archive-date=2020-10-01 }}</ref>

<math>{d \over dx}\log_b(x)={1 \over x\ln (b)}</math>, so <math>{d \over dx}\log_{10}(x)={1 \over x\ln(10)}</math>.

==See also==
*[[Binary logarithm]]
*[[Cologarithm]]
*[[Decibel]]
*[[Logarithmic scale]]
*[[Napierian logarithm]]
*[[Significand]] (also commonly called mantissa)

==Notes==
{{reflist|group=lower-alpha}}

==References==
{{Reflist|refs=
<ref name="Euler_1748">{{cite book |title=Introductio in Analysin Infinitorum (Part 2) |url=https://scholarlycommons.pacific.edu/euler-works/102/ |language=la |author-first1=Leonhard |author-last1=Euler |author-link1=Leonhard Euler |year=1748|publisher=Marcum-Michaelem Bousquet|location=Lausanne|contribution=Chapter 22: Solutio nonnullorum problematum ad Circulum pertinentium|page=304}}</ref>
<ref name="Scherffer_1772">{{cite book |title=Institutionum Analyticarum Pars Secunda de Calculo Infinitesimali Liber Secundus de Calculo Integrali |language=la |author-first=P. Carolo |author-last=Scherffer |publisher=Joannis Thomæ Nob. De Trattnern |date=1772 |page=198 |volume=2 |url=https://books.google.com/books?id=pJlbAAAAcAAJ}}</ref>
<ref name="Hall_1909">{{cite book |title=Trigonometry |volume=Part I: Plane Trigonometry |first1=Arthur Graham |last1=Hall |first2=Fred Goodrich |last2=Frink |date=1909 |chapter=Chapter IV. Logarithms [23] Common logarithms |publisher=[[Henry Holt and Company]] |location=New York |page=31 |url=https://archive.org/stream/planetrigonometr00hallrich#page/n46/mode/1up}}</ref>
<ref name="Hedrick_1913">{{cite book |author-first=Earle Raymond |author-last=Hedrick |url=https://archive.org/details/logarithmictrigo00hedriala |title=Logarithmic and Trigonometric Tables |publisher=[[Macmillan Publishers|Macmillan]] |location=New York, USA |date=1913}}</ref>
}}

== Bibliography ==
* {{AS ref}}
* {{cite book |author-first=Michael |author-last=Möser |title=Engineering Acoustics: An Introduction to Noise Control |publisher=Springer |date=2009 |isbn=978-3-540-92722-8 |page=448 |url=https://books.google.com/books?id=kl2TA53Ysh4C&pg=PA448}}
* {{cite book |author-first1=Andrei Dmitrievich |author-last1=Poliyanin |author-first2=Alexander Vladimirovich |author-last2=Manzhirov |title=Handbook of mathematics for engineers and scientists |publisher=[[CRC Press]] |date=2007 |orig-year=2006-11-27 |isbn=978-1-58488-502-3 |page=9 |url=https://books.google.com/books?id=ge6nk9W0BCcC&pg=PA9}}

{{Authority control}}

[[Category:Logarithms]]