Consider a complex matrix $A$. When does $e^A=e^B$ imply $A=B$?
Is there any general statement that can be made as to when this holds?
It is clearly not true in general, a trivial example being when $A$ and $B$ are diagonal in the same basis but with eigenvalues differing by $2\pi i\mathbb Z$. This is also mentioned in this question.
This answer also seems to provide an answer to this question, but I'm not familiar with the theory of Lie algebras so I'm not sure, and I wouldn't know how to translate it into more elementary statements about matrices (if that is even possible).