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^{t k}

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

Lie algebra of $G L (n)$ is just the entire $M_{n}$
we can immediately exclude non-invertible matrices from being the result of the exponential, because $e^{t M}$ has inverse $e^{- t M}$ , so we already know that non-invertible matrices are not reachable