exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...

, the base is the number b in an expression of the form bⁿ.

Related terms

The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b. It is more commonly expressed as "the nth power of b", "b to the nth power" or "b to the power n". For example, the fourth power of 10 is 10,000 because . The term ''power'' strictly refers to the entire expression, but is sometimes used to refer to the exponent. Radix is the traditional term for ''base'', but usually refers then to one of the common bases: decimal (10), binary (2), hexadecimal (16), or sexagesimal (60). When the concepts of

variable Variable may refer to: * Variable (computer science), a symbolic name associated with a value and whose associated value may be changed * Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...

and constant came to be distinguished, the process of exponentiation was seen to transcend the

algebraic function In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations additi ...

s. In his 1748 ''Introductio in analysin infinitorum'',

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

referred to "base a = 10" in an example. He referred to ''a'' as a "constant number" in an extensive consideration of the function F(''z'') = ''a''^z. First ''z'' is a positive integer, then negative, then a fraction, or rational number.

(1748
Chapter 6: Concerning Exponential and Logarithmic Quantities
of

Introduction to the Analysis of the Infinite ''Introductio in analysin infinitorum'' (Latin: ''Introduction to the Analysis of the Infinite'') is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin and published in 1748, the ''Introducti ...

, translated by Ian Bruce (2013), lk from 17centurymaths.

Roots

When the nth power of b equals a number a, or a = bⁿ, then b is called an " nth root" of a. For example, 10 is a fourth root of 10,000.

Logarithms

The

inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X ...

to exponentiation with base b (when it is

well-defined In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A func ...

) is called the

logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...

to base b, denoted log_b. Thus: :log_''b'' ''a'' = ''n''. For example, log₁₀ 10,000 = 4.

References

{{Reflist Exponentials Mathematical terminology