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.
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 $a = 0$)