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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 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.

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    $\begingroup$ I don't think this is known. If $\log(\pi)=p/q$, then $e^p=\pi^q$, which implies that $\pi$ and $e$ are algebraically dependent. It is widely believed, but not proved, that $\pi$ and $e$ are algebraically independent. $\endgroup$
    – Gerry Myerson
    Oct 25, 2016 at 22:06
  • $\begingroup$ mathoverflow.net/questions/33817/… $\endgroup$
    – Carlo Beenakker
    Oct 26, 2016 at 8:12

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The irrationality of $\log \pi$ is an open problem (see for example this recent paper).

It is expected to be transcendental (page 34 of this slides by Michel Waldschmidt), and in fact this follows from Schanuel's conjecture (this is referenced here, 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.

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    $\begingroup$ Thank you for citing my arXiv paper. In fact, I've investigated the number ln(pi) more closely in my recent paper published in Journal of Analysis and Number Theory 5, p.91 (2017). The title is "Some Transcendence Results from a Harmless Irrationality Theorem". There you will find some nice discussions on irrationality and transcendence of many numbers. Regards, Prof. Fabio M. S. Lima $\endgroup$
    – Dr. Fabio M. S. Lima
    Jul 18, 2021 at 2:36
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    $\begingroup$ The paper is freely available at naturalspublishing.com/files/published/j2s8h5c33e8ci4.pdf It does not settle the question of $\log\pi$ $\endgroup$
    – Gerry Myerson
    Jul 18, 2021 at 3:24
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    $\begingroup$ I note that Natural Sciences Publishing Corporation is on a list of possibly predatory publishers maintained at predatoryjournals.com/publishers . $\endgroup$
    – Gerry Myerson
    Jul 18, 2021 at 3:32
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    $\begingroup$ Hi Gerry Myerson, you are right, the irrationality of ln(pi) remains open. However, according to my Theorem 3, at least 2 of the 3 numbers {pi+e, pi*e, ln(pi)} are TRANSCENDENTAL. I think this gets a sensation that ln(pi) must be transcendental. More work in this direction is needed... $\endgroup$
    – Dr. Fabio M. S. Lima
    Jul 19, 2021 at 12:35
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    $\begingroup$ Let me take this opportunity to say that judge a paper by the name of the journal in which it was published, without reading the paper, is bad. I've chosen JANT by the scope, not by fame! $\endgroup$
    – Dr. Fabio M. S. Lima
    Jul 19, 2021 at 12:43

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