Common logarithm

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.{{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}} The word "mantissa" is often used to describe the part of a floating-point number that represents its 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:

:$\backslash log\_\{10\}(120)\; =\; \backslash log\_\{10\}\backslash left(10^2\; \backslash times\; 1.2\backslash right)\; =\; 2\; +\; \backslash log\_\{10\}(1.2)\; \backslash approx\; 2\; +\; 0.07918.$

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 2.

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

:$\backslash log\_\{10\}(0.012)\; =\; \backslash log\_\{10\}\backslash left(10^\{-2\}\; \backslash times\; 1.2\backslash right)\; =\; -2\; +\; \backslash log\_\{10\}(1.2)\; \backslash approx\; -2\; +\; 0.07918\; =\; -1.92082.$

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:

:$\backslash log\_\{10\}(0.012)\; \backslash approx\; \backslash bar\{2\}\; +\; 0.07918\; =\; -1.92082.$

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 $\backslash bar\{n\}$ is read as "bar {{Mvar|n}}", so that $\backslash bar\{2\}.07918$ is read as "bar 2 point 07918...". An alternative convention is to express the logarithm modulo 10, in which case

:$\backslash log\_\{10\}(0.012)\; \backslash approx\; 8.07918\; \backslash bmod\; 10,$

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) $\backslash log\; b\; =\; 6.51335464$,

$\backslash log\; e\; =\; 8.9054355$.

From the context, it is understood that

$b\; =\; 10^\{6.51335464\}$, the minor radius of the earth ellipsoid

in toise (a large number), whereas

$e\; =\; 10^\{8.9054355-10\}$, the eccentricity of the earth ellipsoid

(a small number).}}

The following example uses the bar notation to calculate 0.012 × 0.85 = 0.0102:

:$\backslash begin\{array\}\{rll\}$

\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}

* 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:

Note that the mantissa is common to all of the {{Math|5 {{times}} 10ⁱ}}. This holds for any positive real number $x$ because

:$\backslash log\_\{10\}\backslash left(x\; \backslash times10^i\backslash right)\; =\; \backslash log\_\{10\}(x)\; +\; \backslash log\_\{10\}\backslash left(10^i\backslash right)\; =\; \backslash log\_\{10\}(x)\; +\; i.$

Since {{Mvar|i}} is a constant, the mantissa comes from $\backslash log\_\{10\}(x)$, which is constant for given $x$. This allows a table of logarithms to include only one entry for each mantissa. In the example of {{Math|5 {{times}} 10ⁱ}}, 0.698 970 (004 336 018 ...) will be listed once indexed by 5 (or 0.5, or 500, etc.).

Image:Slide rule example2.svg 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}}.]]

{{Main|History of logarithms}}

Common logarithms are sometimes also called "Briggsian logarithms" after Henry Briggs, a 17th 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|log₁₀(x)}}. Mathematicians, on the other hand, wrote "{{math|log(x)}}" when they meant {{math|log_e(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 specification recommends that {{math|log_e(x)}} should be {{math|ln(x)}}, while {{math|log₁₀(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.{{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 }}{{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

}}.{{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

}}

File:Logarithm keys.jpg

The numerical value for logarithm to the base 10 can be calculated with the following identities:

: $\backslash log\_\{10\}(x)\; =\; \backslash frac\{\backslash ln(x)\}\{\backslash ln(10)\}\; \backslash quad$ or $\backslash quad\; \backslash log\_\{10\}(x)\; =\; \backslash frac\{\backslash log\_2(x)\}\{\backslash log\_2(10)\}\; \backslash quad$ or $\backslash quad\; \backslash log\_\{10\}(x)\; =\; \backslash frac\{\backslash log\_B(x)\}\{\backslash log\_B(10)\}\; \backslash quad$

using logarithms of any available base $\backslash ,\; B\; ~.$

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

The derivative of a logarithm with a base b is such that{{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 }}

$\{d\; \backslash over\; dx\}\backslash log\_b(x)=\{1\; \backslash over\; x\backslash ln\; (b)\}$, so $\{d\; \backslash over\; dx\}\backslash log\_\{10\}(x)=\{1\; \backslash over\; x\backslash ln(10)\}$.

• Binary logarithm
• Cologarithm
• Decibel
• Logarithmic scale
• Napierian logarithm
• Significand (also commonly called mantissa)

{{reflist|group=lower-alpha}}

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{{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}}

{{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}}

{{cite book |author-first=Earle Raymond |author-last=Hedrick |url=https://archive.org/details/logarithmictrigo00hedriala |title=Logarithmic and Trigonometric Tables |publisher=Macmillan |location=New York, USA |date=1913}}

}}

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Category:Logarithms