TITLE: Are there integers $a, b$ such that $\pi^a = e^b$?
QUESTION [8 upvotes]: Is $\log \pi $ a rational number? That is, are there non-zero integers $a, b$ s.th. $\pi^a = e^b$ ?

REPLY [2 votes]: This is still an open problem. The algebraic independence of $\pi$ and $e$ is unknown, i.e., it is unknown whether there exist algebraic numbers $n_0, n_1, \ldots, n_k$ such that 
$$
n_{0} \pi^{0} + \ldots + n_{k-1} \pi^{k-1} + n_k \pi^{k} = e.
$$