I've found a proof that: $(1)$ if $a\neq1$ is a positive real number than at least one of these numbers are irrational: $a$ or $\ln a$.
I was told that this result is a corollary of Lindemann theorem, that states that if $b$ is algebraic than $e^b$ is transcedental. How does Lindemann theorem implies $(1)?$
I've found $(1)$ to be interesting because if we set $a=e\pi$, than at least one is irrational: $e\pi$ or $\ln \pi$.
You don't need more than $e$ is transcedental. Let $b=\ln a$, and if both of $a,b$ are rational, then $e^b=a$ is rational, and hence $e=a^{1/b}$ is algebraic: If $b=\frac{m}{n}, m,n\in\mathbb Z$, then $e^{m}=a^{n}\in\mathbb Q$.