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Common logarithm

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Mathematical function

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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

Inmathematics, thecommon logarithm (aka "standard logarithm") is thelogarithm with base 10.^[1] It is also known as thedecadic logarithm, thedecimal logarithm and theBriggsian logarithm. The name "Briggsian logarithm" is in honor of the British mathematicianHenry Briggs who conceived of and developed the values for the "common logarithm". Historically, the "common logarithm" was known by its Latin namelogarithmus decimalis^[2] orlogarithmus decadis.^[3]

The mathematical notation for using the common logarithm islog(x),^[4]log₁₀(x),^[5] or sometimesLog(x) with a capitalL;^[a] oncalculators, it is printed as "log",^[6] but mathematicians usually meannatural logarithm (logarithm with basee ≈ 2.71828) rather than common logarithm when writing "log", since the natural logarithm is – contrary to what the name of the common logarithm implies – the most commonly used logarithm in pure math.^[7]

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, andmechanical calculators capable of multiplication were bulky, expensive, and not widely available. Instead,tables of base-10 logarithms were used in science, engineering and navigation—when calculations required greater accuracy than could be achieved with aslide rule. By turning multiplication and division to addition and subtraction, use of logarithms avoided laborious and error-prone paper-and-pencil multiplications and divisions.^[1] Because logarithms were so useful,tables of base-10 logarithms were given in appendices of many textbooks. Mathematical and navigation handbooks included tables of the logarithms oftrigonometric functions as well.^[8] For the history of such tables, seelog table.

Numbers are placed onslide 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 that2 × 3 = 6.

Mantissa and characteristic

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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 samefractional part. The fractional part is known as themantissa.^[b] 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 thecharacteristic, 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:

\log _{10}(120)=\log _{10}\left(10^{2}\times 1.2\right)=2+\log _{10}(1.2)\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.

Negative logarithms

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

\log _{10}(0.012)=\log _{10}\left(10^{-2}\times 1.2\right)=-2+\log _{10}(1.2)\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, calledbar notation, is used:

\log _{10}(0.012)\approx {\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 ${\bar {n}}$ is read as "barn", so that ${\bar {2}}.07918$ is read as "bar 2 point 07918...". An alternative convention is to express the logarithm modulo 10, in which case

\log _{10}(0.012)\approx 8.07918{\bmod {1}}0,

with the actual value of the result of a calculation determined by knowledge of the reasonable range of the result.^[c]

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

{\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 itsantilog (10^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:

Common logarithm, characteristic, and mantissa of powers of 10 times a number
Number	Logarithm	Characteristic	Mantissa	Combined form
n = 5 × 10ⁱ	log₁₀(n)	i = floor(log₁₀(n))	log₁₀(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...	1.698 970...
0.000 005	−5.301 029...	−6	0.698 970...	6.698 970...

Note that the mantissa is common to all of the5 × 10ⁱ. This holds for any positivereal number $x\times 10^{i}$ because

\log _{10}\left(x\times 10^{i}\right)=\log _{10}(x)+\log _{10}\left(10^{i}\right)=\log _{10}(x)+i.

Sincei is a constant, the mantissa comes from $\log _{10}(x)$ , which is constant for given $x {\displaystyle x}$ . This allows atable of logarithms to include only one entry for each mantissa. In the example of5 × 10ⁱ, 0.698 970 (004 336 018 ...) will be listed once indexed by 5 (or 0.5, or 500, etc.).

History

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Main article:History of logarithms

Common logarithms are sometimes also called "Briggsian logarithms" afterHenry Briggs, a 17th century British mathematician. In 1616 and 1617, Briggs visitedJohn Napier atEdinburgh, 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 firstchiliad of his logarithms.

Because base-10 logarithms were most useful for computations, engineers generally simply wrote "log(x)" when they meantlog₁₀(x). Mathematicians, on the other hand, wrote "log(x)" when they meantlog_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 "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, theISO 80000 specification recommends thatlog_e(x) should beln(x), whilelog₁₀(x) should be writtenlg(x), which unfortunately is used for thebase-2 logarithm by CLRS and Sedgwick andThe Chicago Manual of Style.^[10]^[11]^[12]

Numeric value

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The logarithm keys (*log* for base-10 andln for base-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 10 can be calculated with the following identities:^[5]

\log _{10}(x)={\frac {\ln(x)}{\ln(10)}}\quad

\quad \log _{10}(x)={\frac {\log _{2}(x)}{\log _{2}(10)}}\quad

\quad \log _{10}(x)={\frac {\log _{B}(x)}{\log _{B}(10)}}\quad

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

as procedures exist for determining the numerical value forlogarithm basee (seeNatural logarithm § Efficient computation) andlogarithm base 2 (seeAlgorithms for computing binary logarithms).

Derivative

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The derivative of a logarithm with a baseb is such that^[13]

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

Notes

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^The notationLog is ambiguous, as this can also mean the complex natural logarithmicmulti-valued function.
^This use of the wordmantissa stems from an older, non-numerical, meaning: a minor addition or supplement, e.g., to a text.^{[citation needed]} The word was introduced byHenry Briggs.^[9] The word "mantissa" is often used to describe the part of afloating-point number that represents itssignificant digits, although "significand" was the term used for this byIEEE 754, and may be preferred to avoid confusion with logarithm mantissas.
^For example,Bessel, F. W. (1825). "Über die Berechnung der geographischen Längen und Breiten aus geodätischen Vermessungen".Astronomische Nachrichten.331 (8):852–861.arXiv:0908.1823.Bibcode:1825AN......4..241B.doi:10.1002/asna.18260041601.S2CID 118630614.gives (beginning of section 8) $\log b=6.51335464$ , $\log e=8.9054355$ .From the context, it is understood that $b=10^{6.51335464}$ , the minor radius of the earth ellipsoidintoise (a large number), whereas $e=10^{8.9054355-10}$ , the eccentricity of the earth ellipsoid(a small number).