Common logarithm

Before about 1970, hand-held electronic calculators were not yet in widespread use. Because of their utility in saving work in laborious calculations by hand on paper, tables of base-10 logarithms were found in appendices of many books. Base-10 logarithms were then called common logarithms. Such a table of "common logarithms" gave the logarithm 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 no need to include numbers not between 1 and 10, since if one wanted the logarithm of, for example, 120, one would know that

$\log _{10}120=\log _{10}(1.2\times 10^{2})=2+\log _{10}1.2\cong 2+0.079181.$

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 logarithm of 120, is 2. The fractional part of the logarithm, 0.079181, was called the "mantissa" of the logarithm.

Because base-10 logarithms were called "common", and engineers often had occasion to use them, engineers often wrote "$\log(x)$" when they meant $\log _{10}(x)$. Mathematicians, on the other hand, wrote, and still write today, "$\log(x)$" when they mean "$\log _{e}(x)$" (see natural logarithm). 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, has been popularized by the very invention that made the use of "common logarithms" obsolete: electronic calculators.