Logarithm is the exponent or power to which a base must be raised to yield a given number.

An example of a logarithm is as follows. In the expression bx = N, if b is the base and equal to 10 and N a number, equal to 100, then x is equal to 2 and is said to be the logarithm of 100 to the base 10. This is written: log 100 = 2, in which it is understood that log means logarithm to the base 10. The latter is also called a common logarithm. Logarithms that employ the base e, in which e = 2.71828 ... are called natural, or Napierian, logarithms; the notation used is ln, to distinguish natural logarithms from common logarithms (log).

When a common logarithm of a number is written as the sum of an integer and a positive decimal (e.g., 2.3147), the integer--called the characteristic--serves to locate the decimal point in the number, and the decimal--called the mantissa--indicates the digits in the number. The latter are determined from tables of logarithms, which relate mantissas to numbers. When the number is greater than or equal to 1, the characteristic is 1 less than the number of digits to the left of the decimal point; when the number is less than 1, the characteristic is negative and is 1 more than the number of zeros following the decimal point. For example, the number 365.0 has the characteristic 2; the number 0.005 has the characteristic -3.

Excerpt from the Encyclopedia Britannica without permission.