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 ...
, 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 r ...
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
In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is ...
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 ...
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 ...
(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 to exponentiation with base
b (when it is
well-defined) 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 ...
to base
b, denoted log
b. Thus:
:log
''b'' ''a'' = ''n''.
For example, log
10 10,000 = 4.
References
{{Reflist
Exponentials
Mathematical terminology