logarithm

The exponent or index of a number to a specified base – usually 10. For example, the logarithm to the base 10 of 1,000 is 3 because 10³ = 1,000; the logarithm of 2 is 0.3010 because 2 = 10^0.3010. The whole-number part of a logarithm is called the characteristic; the fractional part is called the mantissa.

Before the advent of cheap electronic calculators, multiplication and division could be simplified by being replaced with the addition and subtraction of logarithms.

For any two numbers x and y (where x = b^a and y = b^c), x × y = b^a × b^c = b^{a + c}; hence one would add the logarithms of x and y, and look up this answer in antilogarithm tables.

Tables of logarithms and antilogarithms are available that show conversions of numbers into logarithms, and vice versa. For example, to multiply 6,560 by 980, one looks up their logarithms (3.8169 and 2.9912), adds them together (6.8081), then looks up the antilogarithm of this to get the answer (6,428,800). Natural or Napierian logarithms are to the base e, an irrational number equal to approximately 2.7183.

The principle of logarithms is also the basis of the slide rule. With the general availability of the electronic pocket calculator, the need for logarithms has been reduced. The first log tables (to base e) were published by the Scottish mathematician John Napier in 1614. Base-ten logs were introduced by the Englishman Henry Briggs (1561–1631) and Dutch mathematician Adriaen Vlacq (1600–1667).