Is $e^{(rA)} = (e^{A})^r$ when $r \in \mathbb{R}$ and $A$ is an element of a Banach algebra?

Clearly if $n$ is an integer, then

$e^{(nA)} = e^{A+A \cdots +A} = e^{A}e^{A}\cdots e^{A} = (e^{A})^n$,

where the second equality follows from the Baker-Hausdorff lemma and the fact that [A,A]=0. On the other hand, I think the equality is not generally true when $r \in \mathbb{C}$. But what about the reals?

Many thanks for your thoughts!