The definition of the exponential with integer exponents is straightforward to define: $x^n=\underbrace{x\cdot\ldots\cdot x}_{n-\text{times}}$.
These days I've been thinking about the formal definition of the exponential with real exponents.
My question:
What is the definition of $a^{b}$, where $a,b\in \mathbb R$.
If both $a,b\in \mathbb R$ then we have to define $a^b$ by means of complex functions:
\begin{align*} a^b = e^{bLn(a) } = e^{b\ln(|a|)+i b\arg(a)+2\pi pb i} , \end{align*} $p \in \mathbb Z$. Now the exponential function is defined by the series: \begin{align*} e^x = \sum_{n=0}^\infty \frac{x^n}{n!}. \end{align*}
Noting that $\frac{d}{dx}e^x = e^x >0$ (not as easy as it sounds) we know that $f(x) = e^x$ is monotonically increasing on $\mathbb R$. Thus it has an inverse which is called $\ln$. Now this defines $a^b$. Calculating an approximate value of say $(-2)^\pi$ from this is hard (or at least boring).