TITLE: Is ln(n) transcendental for all integer $n > 1$?
QUESTION [12 upvotes]: Is $\ln(n)$ transcendental for all $n \in \mathbb{N} \setminus \{0, 1\}$? Is the answer even known?

REPLY [4 votes]: Lindemann proved  $e^α$ is transcendental for every non-zero algebraic number $α$. 
Let $n$ be a natural number.
Assume that $\log(n)$ is algebraic. It is non-zero if $n>1$. Thus, for $n>1$, we conclude that $e^{\log n}=n$ is transcendental. Contradiction.