I'm interested in knowing more about the question if $f(\pi)$ is rational or not, where $f$ is some well-known function. For example, $\cos(\pi) =-1$ is rational, while ${e}^{\pi}$ is irrational as shown [here](http://www.wolframalpha.com/input/?i=e%5E%28pi%29++is+irrational+number) by WolframAlpha.


**My question here is**: Is it possible to know if $\log(\pi)$ is rational or not since the $\log$ function is the inverse of the $\exp$ function?


Thank you for any help. 



Is it possible to know if $\log(\pi)$ is irrational or not since the $\log$ function is the inverse of the $\exp$ function?
The irrationality of $\log \pi$ is an open problem (see for example [this](https://arxiv.org/abs/0908.3253) recent paper).


It is expected to be transcendental (page 34 of [this slides](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TNTOpenPbs.pdf) by Michel Waldschmidt), and in fact this follows from Schanuel's conjecture (this is referenced [here](https://arxiv.org/abs/1010.6216), beginning of section 3), which is widely believed to be true.


In particular, to answer the question in the title, it does not follows from the irrationality (or transcendence) of $e^\pi$ by any known argument.


