Number associated with a positive number, being the power to which a third number, called the base, must be raised in order to obtain the given positive number. For example, the logarithm of 100 to the base 10 is 2, written log₁₀ 100=2, since 10²=100. Logarithms of positive numbers using the number 10 as the base are called common logarithms; those using the number e as the base are called natural logarithms or Napierian logarithms (for John Napier). The natural logarithm of a number x is denoted by ln x or simply log x. Since logarithms are exponents, they satisfy all the usual rules of exponents. Consequently, tedious calcula ions such as multiplications and divisions can be replaced by the simpler processes of adding or subtracting the corresponding logarithms. Logarithmic tables are generally used for this purpose.

In mathematics, a number, letter, or algebraic expression written above and to the right of another number, letter, or expression called the base. In the expressions x² and xⁿ, the number 2 and the letter n are the exponents respectively of the base x. The exponent indicates the power to which the base is to be raised. When exponents were first introduced, only positive whole numbers were used, and the exponent indicated how many times the base was to be taken as a factor; e.g., 2⁵=32, or 2•2•2•2•2=32. In advanced algebra, fractions, zero, and negative numbers are also used as exponents. Particular meanings have been assigned to these types of exponents so that they obey the same algebraic rules as does the simpler type of exponent. A fractional exponent such as 1/4 or 1/n indicates the fourth or nth root, respectively, of the base. Any nonzero quantity raised to the zero power equals one; e.g., x⁰=5⁰=(a²+b²)⁰=1. A negative exponent indicates the reciprocal of the quantity; e.g., x^-2 means 1/x². When quantities of the same base are multiplied together, their exponents are added; e.g., x²•x³=x⁵. Note that the base must be the same. When a quantity already containing an exponent is raised to a power, the exponents are multiplied; e.g., (x²)³=x⁶.

e, in mathematics, irrational number occurring widely in mathematics and science, approximately equal to the value 2.71828; it is the base of natural, or Naperian, logarithms. The number e is defined as the limit of the expression (1+1/n)ⁿ as n becomes infinitely large, or In 1873 the French mathematician C. Hermite proved that e was transcendental, i.e., not a root of any algebraic equation; this proof constituted a great contribution to the growth of mathematics. The number e is also known as Euler's number, for Leonhard Euler.