How to define $a^x$?

Question

It's so common that we use the function $f(x)=a^x$. But actually how do we define it? In simple language we can say $a^n$ is the number $a$ multiplied with $a$ $n$ times for any $n$ in $\mathbb{N}$ and $a$ in $\mathbb{R}$. Then what about our $f$?

Answer 1

Let $a$ be positive. Since you know $a^n$ when $n$ is an integer, define $a^{1/n}$ for $n\ne0$ integer as the unique positive number $b$ such that $b^n=a$ (using the continuity and monotonicity of each function $t\mapsto t^n$ on $t\gt0$), then $a^{p/q}$ as $(a^{1/q})^p$ for every integers $p$ and $q$, $q\ne0$ (you can check the result does not depend on $(p,q)$ but only on $p/q$), and finally $a^x$ as the limit of $a^r$ when $r$ rational converges to $x$.

If $a\lt0$ and, say, $n=\frac12$, I seem to remember some Italian guys explained years ago that the result ought to be not real but somewhat more... complex.