Editing Exponentiation (section)

==Real exponents==
For positive real numbers, exponentiation to real powers can be defined in two equivalent ways, either by extending the rational powers to reals by continuity (''{{slink||Limits of rational exponents}}'', below), or in terms of the [[logarithm]] of the base and the [[exponential function]] (''{{slink||Powers via logarithms}}'', below). The result is always a positive real number, and the [[#Identities and properties|identities and properties]] shown above for integer exponents remain true with these definitions for real exponents. The second definition is more commonly used, since it generalizes straightforwardly to [[complex number|complex]] exponents.

On the other hand, exponentiation to a real power of a negative real number is much more difficult to define consistently, as it may be non-real and have several values. One may choose one of these values, called the [[principal value]], but there is no choice of the principal value for which the identity
: <math>\left(b^r\right)^s = b^{r s}</math>
is true; see ''{{slink||Failure of power and logarithm identities}}''. Therefore, exponentiation with a basis that is not a positive real number is generally viewed as a [[multivalued function]].

===Limits of rational exponents===
[[File:Continuity of the Exponential at 0.svg|thumb|The limit of {{math|''e''{{sup|1/''n''}}}} is {{math|1=''e''{{sup|0}} = 1}} when {{mvar|n}} tends to the infinity.]]
Since any [[irrational number]] can be expressed as the [[limit of a sequence]] of rational numbers, exponentiation of a positive real number {{mvar|b}} with an arbitrary real exponent {{mvar|x}} can be defined by [[continuous function|continuity]] with the rule<ref name="Denlinger">{{cite book |title=Elements of Real Analysis |last=Denlinger |first=Charles G. |publisher=Jones and Bartlett |date=2011 |pages=278–283 |isbn=978-0-7637-7947-4}}</ref>
:<math> b^x = \lim_{r (\in \mathbb{Q}) \to x} b^r \quad (b \in \mathbb{R}^+,\, x \in \mathbb{R}),</math>
where the limit is taken over rational values of {{mvar|r}} only. This limit exists for every positive {{mvar|b}} and every real {{mvar|x}}.

For example, if {{math|1=''x'' = {{pi}}}}, the [[non-terminating decimal]] representation {{math|1=''π'' = 3.14159...}} and the [[monotone function|monotonicity]] of the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain <math>b^\pi:</math>
:<math>\left[b^3, b^4\right], \left[b^{3.1}, b^{3.2}\right], \left[b^{3.14}, b^{3.15}\right], \left[b^{3.141}, b^{3.142}\right], \left[b^{3.1415}, b^{3.1416}\right], \left[b^{3.14159}, b^{3.14160}\right], \ldots</math>
So, the upper bounds and the lower bounds of the intervals form two [[sequence (mathematics)|sequences]] that have the same limit, denoted <math>b^\pi.</math>

This defines <math>b^x</math> for every positive {{mvar|b}} and real {{mvar|x}} as a [[continuous function]] of {{mvar|b}} and {{mvar|x}}. See also ''[[Well-defined expression]]''.<ref>{{cite book |chapter=Limits of sequences |chapter-url={{Google books |ecTsDAAAQBAJ |page=154 |plainurl=yes}} |title=Analysis I |series=Texts and Readings in Mathematics |year=2016 |last1=Tao |first1=Terence |volume=37 |pages=126–154 |isbn=978-981-10-1789-6 |doi=10.1007/978-981-10-1789-6_6}}</ref>

===Exponential function===
{{Main|Exponential function}}
The ''exponential function'' may be defined as <math>x\mapsto e^x,</math> where <math>e\approx 2.718</math> is [[Euler's number]], but to avoid [[circular reasoning]], this definition cannot be used here. Rather, we give an independent definition of the exponential function <math>\exp(x),</math> and of <math>e=\exp(1)</math>, relying only on positive integer powers (repeated multiplication). Then we sketch the proof that this agrees with the previous definition: <math>\exp(x)=e^x.</math>

There are [[characterizations of the exponential function|many equivalent ways to define the exponential function]], one of them being
: <math>\exp(x) = \lim_{n\rightarrow\infty} \left(1 + \frac{x}{n}\right)^n.</math>

One has <math>\exp(0)=1,</math> and the ''exponential identity'' (or multiplication rule) <math>\exp(x)\exp(y)=\exp(x+y)</math> holds as well, since
: <math>\exp(x)\exp(y) = \lim_{n\rightarrow\infty} \left(1 + \frac{x}{n}\right)^n\left(1 + \frac{y}{n}\right)^n = \lim_{n\rightarrow\infty} \left(1 + \frac{x+y}{n} + \frac{xy}{n^2}\right)^n,</math>
and the second-order term <math>\frac{xy}{n^2}</math> does not affect the limit, yielding <math>\exp(x)\exp(y) = \exp(x+y)</math>.

Euler's number can be defined as <math>e=\exp(1)</math>. It follows from the preceding equations that <math>\exp(x)=e^x</math> when {{mvar|x}} is an integer (this results from the repeated-multiplication definition of the exponentiation). If {{mvar|x}} is real, <math>\exp(x)=e^x</math> results from the definitions given in preceding sections, by using the exponential identity if {{mvar|x}} is rational, and the continuity of the exponential function otherwise.

The limit that defines the exponential function converges for every [[complex number|complex]] value of {{mvar|x}}, and therefore it can be used to extend the definition of <math>\exp(z)</math>, and thus <math>e^z,</math> from the real numbers to any complex argument {{mvar|z}}. This extended exponential function still satisfies the exponential identity, and is commonly used for defining exponentiation for complex base and exponent.

===Powers via logarithms===
The definition of {{math|''e''<sup>''x''</sup>}} as the exponential function allows defining {{math|''b''<sup>''x''</sup>}} for every positive real numbers {{mvar|b}}, in terms of exponential and [[logarithm]] function. Specifically, the fact that the [[natural logarithm]] {{math|ln(''x'')}} is the [[inverse function|inverse]] of the exponential function {{math|''e''<sup>''x''</sup>}} means that one has
: <math>b = \exp(\ln b)=e^{\ln b}</math>
for every {{math|''b'' > 0}}. For preserving the identity <math>(e^x)^y=e^{xy},</math> one must have
: <math>b^x=\left(e^{\ln b} \right)^x = e^{x \ln b}</math>

So, <math>e^{x \ln b}</math> can be used as an alternative definition of {{math|''b''<sup>''x''</sup>}} for any positive real {{mvar|b}}. This agrees with the definition given above using rational exponents and continuity, with the advantage to extend straightforwardly to any complex exponent.