LOGARITHM (from Gr. [Greek: logos], word, ratio, and [Greek: arithmos], number), in mathematics, a word invented by John Napier to denote a particular class of function discovered by him, and which may be defined as follows: if a, x, m are any three quantities satisfying the equation a^x = m, then a is called the base, and x is said to be the logarithm of m to the base a. This relation between x, a, m, may be expressed also by the equation x = log(a) m. _Properties._--The principal properties of logarithms are given by the equations log(a) (mn) = log(a)m + log(a)n, log(a)(m/n) = log(a)m - log(a)n, log(a)m^(r) = r log(a)m, log(a)[root r]m = (1/r)log(a)m, which may be readily deduced from the definition of a logarithm. It follows from these equations that the logarithm of the product of any number of quantities is equal to the sum of the logarithms of the quantities, that the logarithm of the quotient of two quantities is equal to the logarithm of the numerator diminished by the logarithm of the denominator, that the logarithm of the rth power of a quantity is equal to r times the logarithm of the quantity, and that the logarithm of the rth root of a quantity is equal to (1/r)th of the logarithm of the quantity. Logarithms were originally invented for the sake of abbreviating arithmetical calculations, as by their means the operations of multiplication and division may be replaced by those of addition and subtraction, and the operations of raising to powers and extraction of roots by those of multiplication and division. For the purpose of thus simplifying the operations of arithmetic, the base is taken to be 10, and use is made of tables of logarithms in which the values of x, the logarithm, corresponding to values of m, the number, are tabulated. The logarithm is also a function of frequent occurrence in analysis, being regarded as a known and recognized function like sin x or tan x; but in mathematical investigations the base generally employed is not 10, but a certain quantity usually denoted by the letter e, of value 2.71828 18284.... Thus in arithmetical calculations if the base is not expressed it is understood to be 10, so that log m denotes log10 m; but in analytical formulae it is understood to be e. The logarithms to base 10 of the first twelve numbers to 7 places of decimals are log 1 = 0.0000000 log 5 = 0.6989700 log 9 = 0.9542425 log 2 = 0.3010300 log 6 = 0.7781513 log 10 = 1.0000000 log 3 = 0.4771213 log 7 = 0.8450980 log 11 = 1.0413927 log 4 = 0.6020600 log 8 = 0.9030900 log 12 = 1.0791812 The meaning of these results is that 1 = 10^0, 2 = 10^(0.3010300), 3 = 10^(0.4771213), ... 10 = 10^1, 11 = 10^(1.0413927), 12 = 10^(1.0791812).