Common logarithm

Inmathematics,thecommon logarithmis thelogarithmwith base 10.^[1]It is also known as thedecadic logarithmand as thedecimal logarithm,named after its base, orBriggsian logarithm,afterHenry Briggs,an English mathematician who pioneered its use, as well asstandard logarithm.Historically, it was known aslogarithmus decimalis^[2]orlogarithmus decadis.^[3]It is indicated by $log(x)$ ,^[4] $log 10 (x)$ ,^[5]or sometimes $Log(x)$ with a capital $L$ ;^{[note 1]}oncalculators,it is printed as "log", but mathematicians usually meannatural logarithm(logarithm with base e ≈ 2.71828) rather than common logarithm when writing "log". To mitigate this ambiguity, theISO 80000 specificationrecommends that $log 10 (x)$ should be written $lg(x)$ ,and $log e (x)$ should be $ln(x)$ .

Before the early 1970s, handheld electronic calculators were not available, andmechanical calculatorscapable of multiplication were bulky, expensive and not widely available. Instead,tablesof 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,tablesof base-10 logarithms were given in appendices of many textbooks. Mathematical and navigation handbooks included tables of the logarithms oftrigonometric functionsas well.^[6]For the history of such tables, seelog 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 themantissa.^{[note 2]}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

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 "bar $n$ ",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.^{[note 3]}

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 the $5 \times 10 i$ .This holds for any positivereal number $x$ because

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

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

Numbers are placed onslide rulescales 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 \times 3 = 6

.

History

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

Because base-10 logarithms were most useful for computations, engineers generally simply wrote " $log(x)$ "when they meant $log 10 (x)$ .Mathematicians, on the other hand, wrote " $log(x)$ "when they meant $log e (x)$ 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 " $ln(x)$ "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.

Numeric value

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

or

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

or

\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 base $e$ (seeNatural logarithm § Efficient computation) andlogarithm base 2(seeAlgorithms for computing binary logarithms).

Derivative

The derivative of a logarithm with a basebis such that^[7]

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

Notes

^The notation $Log$ is ambiguous, as this can also mean the complex natural logarithmicmulti-valued function.
^This use of the wordmantissastems from an older, non-numerical, meaning: a minor addition or supplement, e.g., to a text. Nowadays, the wordmantissais generally used to describe the fractional part of afloating-pointnumber on computers, though the recommended^{[by whom?]}term issignificand.
^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 ellipsoid intoise(a large number), whereas $e=10^{8.9054355-10}$ ,the eccentricity of the earth ellipsoid (a small number).

References

^^a ^bHall, Arthur Graham; Frink, Fred Goodrich (1909). "Chapter IV. Logarithms [23] Common logarithms".Trigonometry.Vol. Part I: Plane Trigonometry. New York:Henry Holt and Company.p. 31.
^Euler, Leonhard;Speiser, Andreas;du Pasquier, Louis Gustave;Brandt, Heinrich;Trost, Ernst (1945) [1748].Speiser, Andreas(ed.).Introductio in Analysin Infinitorum (Part 2).1 (in Latin). Vol. 9.B.G. Teubner.{{cite book}}:|work=ignored (help)
^Scherffer, P. Carolo (1772).Institutionum Analyticarum Pars Secunda de Calculo Infinitesimali Liber Secundus de Calculo Integrali(in Latin). Vol. 2. Joannis Thomæ Nob. De Trattnern. p. 198.
^"Introduction to Logarithms".www.mathsisfun.com.Retrieved2020-08-29.
^^a ^bWeisstein, Eric W."Common Logarithm".mathworld.wolfram.com.Retrieved2020-08-29.
^Hedrick, Earle Raymond (1913).Logarithmic and Trigonometric Tables.New York, USA:Macmillan.
^"Derivatives of Logarithmic Functions".Math24.2021-04-14.Archivedfrom the original on 2020-10-01.

Bibliography

Abramowitz, Milton;Stegun, Irene Ann,eds. (1983) [June 1964].Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications.ISBN 978-0-486-61272-0.LCCN 64-60036.MR 0167642.LCCN 65-12253.
Möser, Michael (2009).Engineering Acoustics: An Introduction to Noise Control.Springer. p. 448.ISBN 978-3-540-92722-8.
Poliyanin, Andrei Dmitrievich; Manzhirov, Alexander Vladimirovich (2007) [2006-11-27].Handbook of mathematics for engineers and scientists.CRC Press.p. 9.ISBN 978-1-58488-502-3.

[6] The notation $Log$ is ambiguous, as this can also mean the complex natural logarithmicmulti-valued function.

[8] This use of the wordmantissastems from an older, non-numerical, meaning: a minor addition or supplement, e.g., to a text. Nowadays, the wordmantissais generally used to describe the fractional part of afloating-pointnumber on computers, though the recommended^{[by whom?]}term issignificand.

[9] 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 ellipsoid intoise(a large number), whereas $e=10^{8.9054355-10}$ ,the eccentricity of the earth ellipsoid (a small number).

[Hall_1909-1] Hall, Arthur Graham; Frink, Fred Goodrich (1909). "Chapter IV. Logarithms [23] Common logarithms".Trigonometry.Vol. Part I: Plane Trigonometry. New York:Henry Holt and Company.p. 31.

[Euler_1748-2] Euler, Leonhard;Speiser, Andreas;du Pasquier, Louis Gustave;Brandt, Heinrich;Trost, Ernst (1945) [1748].Speiser, Andreas(ed.).Introductio in Analysin Infinitorum (Part 2).1 (in Latin). Vol. 9.B.G. Teubner.{{cite book}}:|work=ignored (help)

[Scherffer_1772-3] Scherffer, P. Carolo (1772).Institutionum Analyticarum Pars Secunda de Calculo Infinitesimali Liber Secundus de Calculo Integrali(in Latin). Vol. 2. Joannis Thomæ Nob. De Trattnern. p. 198.

[4] "Introduction to Logarithms".www.mathsisfun.com.Retrieved2020-08-29.

[:0-5] Weisstein, Eric W."Common Logarithm".mathworld.wolfram.com.Retrieved2020-08-29.

[Hedrick_1913-7] Hedrick, Earle Raymond (1913).Logarithmic and Trigonometric Tables.New York, USA:Macmillan.

[10] "Derivatives of Logarithmic Functions".Math24.2021-04-14.Archivedfrom the original on 2020-10-01.

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