# Common logarithm

Common logarithm
The common logarithm.

The common logarithm is the logarithm with base 10. It is also known as the decadic logarithm, named after its base. It is indicated by log10(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 mathematicians usually mean natural logarithm rather than common logarithm when they write "log". To mitigate this ambiguity the ISO specification is that log10(x) should be lg (x) and loge(x) should be ln (x).

## Uses

Before the early 1970s, hand-held electronic calculators were not yet in widespread use. Because of their utility in saving work in laborious multiplications and divisions with pen and paper, tables of base 10 logarithms were found 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 if one wanted the logarithm of, for example, 120, one would know that

$\log_{10}120=\log_{10}(10^2\times 1.2)=2+\log_{10}1.2\approx2+0.079181.$

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. (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.) 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.

Numbers between (and excluding) 0 and 1 have negative logarithms. For example,

$\log_{10}0.012=\log_{10}(10^{-2}\times 1.2)=-2+\log_{10}1.2\approx-2+0.079181=-1.920819$

To avoid the need for separate tables to convert positive and negative logarithms back to their original numbers, a bar notation is used:

$\log_{10}0.012\approx-2+0.079181=\bar{2}.079181$

The bar over the characteristic indicates that it is negative whilst the mantissa remains positive.

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) − characteristic
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 5×10i. A table of logarithms will have a single indexed entry for the same mantissa. 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..

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

$\begin{array}{rll} \text{As found above,} &\log_{10}0.012\approx\bar{2}.079181 \\ \text{Since}\;\;\log_{10}0.85&=\log_{10}(10^{-1}\times 8.5)=-1+\log_{10}8.5&\approx-1+0.929419=\bar{1}.929419\;, \\ \log_{10}(0.012\times 0.85) &=\log_{10}0.012+\log_{10}0.85 &\approx\bar{2}.079181+\bar{1}.929419 \\ &=(-2+0.079181)+(-1+0.929419) &=-(2+1)+(0.079181+0.929419) \\ &=-3+1.008600 &=-2+0.008600\;^* \\ &\approx\log_{10}(10^{-2})+\log_{10}(1.02) &=\log_{10}(0.01\times 1.02) \\ &=\log_{10}(0.0102) \end{array}$

* This step makes the mantissa between 0 and 1, so that its antilog (10mantissa) can be looked up.

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.

## History

Common logarithms are sometimes also called Briggsian logarithms after Henry Briggs, a 17th-century British mathematician.

Because base 10 logarithms were most useful for computations, engineers generally wrote "log(x)" when they meant log10(x). Mathematicians, on the other hand, wrote "log(x)" when they mean loge(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 ironically, that 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 identity.

$\log_{10}(x) = \frac{\ln(x)}{\ln(10)} \qquad \text{ or } \qquad \log_{10}(x) = \frac{\log_2(x)}{\log_2(10)}$

as procedures exist for determining the numerical value for logarithm base e and logarithm base 2.

## References

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### Look at other dictionaries:

• common logarithm — n. Math. a logarithm having 10 for its base …   English World dictionary

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• common logarithm — logarithm with the number ten as its base (Mathematics) …   English contemporary dictionary

• common logarithm — Math. a logarithm having 10 as the base. Also called Briggsian logarithm. Cf. natural logarithm. [1890 95] * * * …   Universalium

• common logarithm — com′mon log′arithm n. math. a logarithm having 10 as the base Compare natural logarithm • Etymology: 1890–95 …   From formal English to slang

• common logarithm — noun a logarithm to the base 10 • Hypernyms: ↑logarithm, ↑log …   Useful english dictionary

• common logarithm — noun Date: 1849 a logarithm whose base is 10 …   New Collegiate Dictionary

• common logarithm — noun a logarithm to the base 10 …   English new terms dictionary

• common logarithm — /kɒmən ˈlɒgərɪðəm/ (say komuhn loguhridhuhm) noun Mathematics a logarithm using 10 as the base …   Australian English dictionary

• Logarithm — The graph of the logarithm to base 2 crosses the x axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3) …   Wikipedia