Let $e^A$ denote the matrix exponential, for $n \times n$ matrices over $\mathbb{C}$. I am trying to find for which $B$ there exists a solution $A$ to the equation $e^A = B$.
Clearly a necessary condition is for $B$ to be invertible using the fact that $$\text{det}(e^A) = e^{\text{tr}(A)}$$
I suspect that this is a sufficient condition too, but I am not sure how to prove it. I considered using Jordan canonical form to split $B$ into a sum of diagonalizable and nilpotent matrices, but do not know how to proceed. Thank you.