Logarithm.

Logarithm. Logarithm, in mathematics, the exponent or power to which a stated number, called the base, is raised to yield a specific number. For example, in the expression 102 = 100, the logarithm of 100 to the base 10 is 2. This is written log10100 = 2. Logarithms were originally invented to help simplify the arithmetical processes of multiplication, division, expansion to a power, and extraction of a root, but they are now used for a variety of purposes in pure and applied mathematics. The first tables of logarithms were published independently by the Scottish mathematician John Napier in 1614 and the Swiss mathematician Justus Byrgius in 1620. The first table of common logarithms was compiled by the English mathematician Henry Briggs. Common logarithms use the number 10 as the base number. A system of logarithms often employed uses the transcendental number e as a base; they are called natural logarithms. The method of logarithms can be illustrated by considering a sequence of powers of the number 2: 21, 22, 23, 24, 25, and 26, corresponding to the sequence of numbers 2, 4, 8, 16, 32, and 64. The exponents 1, 2, 3, 4, 5, and 6 are the logarithms of these numbers to the base 2. To multiply any number in this sequence by any other number in the series it is only necessary to add the logarithms of the numbers, then find the antilogarithm of the sum of the logarithms, which is equal to the base number raised to the power of the sum. Thus, to multiply 16 by 4, first note that the logarithm of 16 is 4, and the logarithm of 4 is 2. The sum of the logarithms 4 and 2 is equal to 6, and the antilogarithm of 6 is 64, which is the product desired. In division the logarithms are subtracted. To divide 32 by 8 subtract 3 from 5, giving 2, which is the logarithm of the quotient, 4. To expand a number to any power, multiply the logarithm by the power desired, and take the antilogarithm of the product. Thus, to find 43: log 2 4 = 2; 3 × 2 = 6; antilog 6 = 64, which is the third power of 4. Roots are extracted by dividing the logarithm by the desired root. To find the fifth root of 32: log232 = 5; 5 ÷ 5 = 1; antilog 1 = 2, which is the fifth root of 32. The problem in constructing a table of logarithms is to make the intervals between successive entries sufficiently small. In the above example, where the entries are the powers 2, 4, 8, and so on, the entries are too far apart to be useful in multiplying any larger numbers. By advanced mathematical processes, the logarithm of any number to any base can be calculated, and exhaustive tables of logarithms have been prepared. Each logarithm consists of a whole number and a decimal fraction, called respectively the characteristic and the mantissa. In the common system of logarithms, which has the base 10, the logarithm of the number 7 has the characteristic 0 and the mantissa .84510 (correct to five decimal places) and is written 0.84510. The logarithm of the number 70 is 1.84510; and the logarithm of the number 700 is 2.84510. The logarithm of the number .7 is -0.15490, which is sometimes written 9.84510-10 for convenience in calculation. Logarithm tables have been replaced by electronic calculators and computers with logarithmic functions. Contributed By: James Singer Reviewed By: J. Lennart Berggren Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.