Common Logarithm - Uses

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:

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. 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,

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

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.

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