number theory - Does there exist any positive integer $n$ such that $e^n$ is an integer (to show $log 2$ is irrational)?

Does there exist any positive integer $n$ such that $e^n$ is an integer ?

I was in particular trying to prove $\log 2$ is irrational; now if it is rational, then there are relatively prime integers $p,q$ both positive such that $\log2 =p/q$ that is $e^p=2^q$ is an integer. I wanted to reach a contradiction.