Let $A$ be a Metzler matrix, i.e. a real matrix (not necessarily symmetric) whose off-diagonal elements are all non-negative. Then, for $t\ge 0$, the matrix exponential $\exp(At)$ will have all non-negative elements.

My question is, given a single element of $\exp(At)$, is it always a log-concave function of $t$, for $t\ge 0$? That is, is the function
$$
f(t) = \log\Big(\big(\exp(At)\big)_{ij}\Big)
$$
a concave function of $t$ for $t\ge 0$? Note that in this expression $\exp$ is a matrix exponential, but $\log$ is just an ordinary logarithm of a positive number.

By "concave" I mean convex downward, i.e. with decreasing slope.

This should be a simple case of finding the second derivative and showing that it can't be positive, but I haven't seen a way to do that. I haven't been able to find a counterexample either.