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.
$\blacksquare$