I have $a, b \in \mathbb{R} \setminus \mathbb{Q}$, I want to know the result of $a^b$, but I don't know exact $a, b$ because I write them in numeric form. My question is how many digits of $a, b$ have to I know to get $a^b$ with a fixed number of significant digits?
For example $a = \pi, b = e$, I want $\pi ^ e$ with 10 significant digits: how many digits at least of $\pi$ and $e$ I need?