Lŏg'a-rĭthm, n.

n. [Fr. logarithme, Sp. & It. logaritmo, from Gr. λόγος, word, account, proportion, and ἀριθμός, number.] (Math.) One of a class of auxiliary numbers, devised by Napier to abridge arithmetical calculations, by the use of addition and subtraction in place of multiplication and division, the relation of these to common numbers being 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  10000 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 10² = 100, and 3 is the logarithm of 1000, because 10³ = 1000.

Arithmetical complement of a logarithm, the difference between a logarithm and the number ten. See COMPLEMENT.—Common logarithms, or Briggs'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 or 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 Gauss, and are of great service in many astronomical computations.—Hyperbolic or Naperian logarithms, those logarithms of which the base is 2.718281;—so called from Lord Napier, the inventor of logarithms.