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 log_e(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 (10^mantissa) 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 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 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.

• Natural logarithm#Numerical value
• Algorithms for computing binary logarithms

See also

• History of logarithms

References

• Michael Möser: Engineering Acoustics: An Introduction to Noise Control. Springer 2009, ISBN 9783540927228, p. 448 (restricted online copy at Google Books)
• A. D. Poliyanin, A. V. Manzhirov: Handbook of mathematics for engineers and scientists. CRC Press 2007, ISBN 9781584885023, p. 9 (restricted online copy at Google Books)

External links

• Briggsian logarithms on PlanetMath includes a detailed example of using logarithm tables