Common logarithm

Before the early 1970s, 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 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

<math>\log_{10}120=\log_{10}(10^2\times 1.2)=2+\log_{10}1.2\cong2+0.079181.</math>

The very last number -- 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. (For a more modern use of the word "mantissa", see floating point.)

Common logarithms are sometimes also called Briggsian logarithms after William Briggs[?], a 17th-century British mathematician.

Because base-10 logarithms were called "common", and engineers often had occasion to use them, engineers often wrote "<math>\log(x)</math>" when they meant <math>\log_{10}(x)</math>. Mathematicians, on the other hand, wrote "<math>\log(x)</math>" when they mean "<math>\log_e(x)</math>" (see natural logarithm). Today, both notations are found among mathematicians. 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 "<math>\ln(x)</math>" when the natural logarithm is intended, has been further popularized by the very invention that made the use of "common logarithms" obsolete: electronic calculators.