en.wikipedia.org/wiki/Logarithm_of_a_matrix#Existence mentions it always exists for all invertible complex matrices. But the real condition is more complicated. Notable counter example: -1 cannot be reached by any real $e_{tk}$.

The Lie algebra exponential covering problem can be seen as a generalized version of this problem, because

- Lie algebra of $GL(n)$ is just the entire $M_{n}$
- we can immediately exclude non-invertible matrices from being the result of the exponential, because $e_{tM}$ has inverse $e_{−tM}$, so we already know that non-invertible matrices are not reachable