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The definition of exponentiation is commonly defined as:

$$x^n:=\underbrace{x\cdot x\cdot x\cdot ...\cdot x}_\text{n times}$$

when $n \in \mathbb N $.

Given that, what is the definition when $n \notin \mathbb N $ ? for example when $n$ is a real number, what about complex numbers, irrational numbers, matrices? does the definition extends or is it a completely different one, if so, what is it for every case? I also would like to know who defined exponentiation for these cases and in which paper.

I read for a non-reliable source that the actual definition of a number exponentiated itself is $x^x := \exp(x \ln(x))$ is this an extension of the real exponent definition, or am I missing something and this is different than an exponent?

In 702414 Gyu Eun Lee explains what exponentiation really means, I am not looking for that, I am looking for the strict definition.

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    $\begingroup$ There are many possible definitions. There is no single "right" one. A very common very general definition is to use the power series definition, $e^x = 1 + x + x^2/2! + x^3/3! + \cdots$. This also works for complex numbers, matrices, and elsewhere. A slightly different question is to show that the various definitions are all equivalent, but this is well-trodden on this site. $\endgroup$
    – davidlowryduda
    Jun 21, 2021 at 18:00
  • $\begingroup$ The general way to extend exponentiation to other fields (although I'm not sure whether it relates to the actual definitions) is to say $a^b = \exp(b \ln(a)),$ where you can define the natural exponential and logarithm by their limit definitions or power series $\endgroup$ Jun 21, 2021 at 18:00
  • $\begingroup$ Welcome to MSE! Exponentiation $a^b$ is defined to be $e^{b \ln(a)}$. Then, depending on which book you read, there are a handful of definitions for what $e^x$ and $\ln(x)$ should be (though the power series definitions are quite good, since you can plug any old complex number, square matrix, etc into the power series and get something that might be meaningful). You can read more here $\endgroup$ Jun 21, 2021 at 18:00
  • $\begingroup$ Note that different definitions work for different domains (in both base and exponent) - but the different definitions fortunately tend to agree when several are applicable . For example $a^b=\exp(b\ln a))$ fails when $a=0$ which even makes many believe that $0^0$ is undefined when it clearly is $=1$ (because the number of maps from the empty set to the empty set is one) $\endgroup$ Jun 21, 2021 at 18:03
  • $\begingroup$ I also would like to know who defined exponentiation for these cases and in which paper. --- FYI, elementary things like this from well before the last hundred years or so often have such complicated histories (involving parallel and independent paths, rediscoveries, different interpretations from how we presently think about things, issues over whether to count limited readership locally published books/papers or private letters or lectures given but not formally recorded or . . .) that the question is not "well posed", let alone having a known answer. $\endgroup$ Jun 21, 2021 at 18:04

1 Answer 1

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  • $x^x = \exp(x\cdot\ln(x))$ is not the extension of exponentiation,

it's the consequence of this : $f(f^{-1}(x)) = x$ Where the input is $x^x$ instead of just $x$, and $f(x) = \exp(x) = e^x, f^{-1}(x) = \ln(x)$.

  • If you want to compute the value of an exponential function $a^{\textstyle x}$ at some $x = x_0$ where $x_0 \notin \mathbb{Z}$ you would have to compute it manually via what's called "Taylor Series" or "Maclaurin Series" (which is Taylor series centered at x = 0), search for this series on Google on your own.

  • For some $x < 0, x\in \mathbb{Z}$, $a^{x} = \dfrac{1}{a^{x}}$ by consequence of the basic definition of exponention $\displaystyle a^x = \prod_{k=1}^x a$

  • For $x = 0$, the definition of $a^x$ is that $a^x = 1$ for any real number $a$ (including $a = 0$)

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