In
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
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 ...
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 and
constant came to be distinguished, the process of exponentiation was seen to transcend the
algebraic functions.
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.
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 ...
(1748
Chapter 6: Concerning Exponential and Logarithmic Quantities
of Introduction to the Analysis of the Infinite, 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 ...
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 fun ...
) 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 of ...
to base
b, denoted log
b. Thus:
:log
''b'' ''a'' = ''n''.
For example, log
10 10,000 = 4.
References
{{Reflist
Exponentials
Mathematical terminology