Schanuel's Conjecture Implies Transcendence of Log Pi

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Theorem

Let Schanuel's Conjecture be true.


Then the logarithm of $\pi$ (pi):

$\ln \pi$

is transcendental.


Proof

Assume the truth of Schanuel's Conjecture.

From Schanuel's Conjecture Implies Algebraic Independence of Pi and Log of Pi over the Rationals, $\ln \pi$ and $\pi$ are algebraically independent over the rational numbers $\Q$.

Therefore, if Schanuel's Conjecture holds, $\ln \pi$ must be transcendental.

$\blacksquare$