**Log"a*rithm** (?), n. [Gr. word, account, proportion + number: cf. F. *logarithme*.] Math.

One of a class of auxiliary numbers, devised by John Napier, of Merchiston, Scotland (1550-1617), to abridge arithmetical calculations, by the use of addition and subtraction in place of multiplication and division. The relation of *logarithms* to common numbers is that of numbers in an arithmetical series to corresponding numbers in a geometrical series, so that sums and differences of the former indicate respectively products and quotients of the latter; thus

0 1 2 3 4 Indices or logarithms
1 10 100 1000 10,000 Numbers in geometrical progression

Hence, the logarithm of any given number is the exponent of a power to which another given invariable number, called the *base*, must be raised in order to produce that given number. Thus, let 10 be the base, then 2 is the logarithm of 100, because 10^{2} = 100, and 3 is the logarithm of 1,000, because 10^{3} = 1,000.

Arithmetical complement of a logarithm, the difference between a logarithm and the number ten. -- Binary logarithms. See under Binary. -- Common logarithms, or Brigg's logarithms, logarithms of which the base is 10; -- so called from Henry *Briggs*, who invented them. -- Gauss's logarithms, tables of logarithms constructed for facilitating the operation of finding the logarithm of the sum of difference of two quantities from the logarithms of the quantities, one entry of those tables and two additions or subtractions answering the purpose of three entries of the common tables and one addition or subtraction. They were suggested by the celebrated German mathematician Karl Friedrich *Gauss* (died in 1855), and are of great service in many astronomical computations. -- Hyperbolic, or Napierian, logarithms <-- usually called 'natural logarithms' --> , those logarithms (devised by John Speidell, 1619) of which the base is 2.7182818; -- so called from *Napier*, the inventor of logarithms. -- Logistic or Proportional logarithms., See under Logistic.

© Webster 1913.