Log 10
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In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and also as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered its use. It is indicated by log_{10}(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 log_{10}(x) should be lg (x) and log_{e}(x) should be ln (x).
Uses
Before the early 1970s, handheld electronic calculators were not yet in widespread use. Due to their utility in saving work in laborious multiplications and divisions with pen and paper, tables of base 10 logarithms were given 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 the logarithms of larger numbers were easily calculated.
For example, the logarithm of 120 is given by:
- $\backslash log\_\{10\}120=\backslash log\_\{10\}(10^2\backslash times\; 1.2)=2+\backslash log\_\{10\}1.2\backslash approx2+0.079181.$
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.^{[note 1]} 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,
- $\backslash log\_\{10\}0.012=\backslash log\_\{10\}(10^\{-2\}\backslash times\; 1.2)=-2+\backslash log\_\{10\}1.2\backslash approx-2+0.079181=-1.920819$
To avoid the need for separate tables to convert positive and negative logarithms back to their original numbers, a bar notation is used:
- $\backslash log\_\{10\}0.012\backslash approx-2+0.079181=\backslash bar\{2\}.079181$
The bar over the characteristic indicates that it is negative whilst the mantissa remains positive.
number | logarithm | characteristic | mantissa | combined form |
---|---|---|---|---|
n (= 5 × 10^{i}) | log_{10}(n) | i (= floor(log_{10}(n)) ) | log_{10}(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 5×10^{i}. 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:
- $\backslash begin\{array\}\{rll\}$
\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.
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 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 identity.
- $\backslash log\_\{10\}(x)\; =\; \backslash frac\{\backslash ln(x)\}\{\backslash ln(10)\}\; \backslash qquad\; \backslash text\{\; or\; \}\; \backslash qquad\; \backslash log\_\{10\}(x)\; =\; \backslash frac\{\backslash log\_2(x)\}\{\backslash log\_2(10)\}$
as procedures exist for determining the numerical value for logarithm base e and logarithm base 2.
See also
Notes
References
- Michael Möser: Engineering Acoustics: An Introduction to Noise Control. Springer 2009, ISBN 978-3-540-92722-8, p. 448 (restricted online copy, p. 448, at Google Books)
- A. D. Poliyanin, A. V. Manzhirov: Handbook of mathematics for engineers and scientists. CRC Press 2007, ISBN 978-1-58488-502-3, p. 9 (restricted online copy, p. 9, at Google Books)
External links
- Template:Planetmath reference includes a detailed example of using logarithm tables