From Lindemann–Weierstrass theorem, it is known that $e^a$ for non-zero algebraic $a$ is always transcendental. But if $a$ is transcendental, is the opposite ($e^a \in \mathbb A$) always true?
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No. There are uncountably many transcendental $a\in\Bbb R$, each with a different $e^a$, of which only countably many are algebraic.
J.G.
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And a famous example is Gelfond's constant. – dke Apr 14 '22 at 12:36