TITLE: Why is exponentiation defined as $x^y=e^{\ln(x)\cdot y}$?
QUESTION [5 upvotes]: There are many curves that extend integer exponentiation to larger domains, so why was this one chosen?

REPLY [2 votes]: What you want is that $x^y$ be continuous both in $x$ and $y$, and to coincide with the (real branch of) $x^y$ for rational $y$. For simplicity, apply natural logarithms to the relevant equations.