For any field $k$ of characteristic $p$ the sets $$\left\{g \in GL_n(k) \ \middle| \ g^p = 1\right\} \qquad \text{and} \qquad \left\{x \in \mathbb M_n(k) \ \middle| \ x^p = 0\right\}$$ are in bijection. The maps either way are given by the matrix exponential and matrix logarithm. I'd like to know if this bijection between $p$-unipotent elements of the group and $p$-nilpotent elements of the Lie algebra restricts to a similar bijection for certain closed subgroups of $GL_n(k)$.
It's not true in general, but it's not hard to show that for $SL$, $SO$, and $Sp$ the matrix exponential restricts appropriately. Also if $A$ is an algebra and we look at algebra automorphisms $Aut(A) \subseteq GL(A)$ then the exponential map takes derivations $Der(A) \subseteq End(A)$ to automorphisms.
Where I'm stuck is I can't figure out if the logarithm restricts as well, or equivalently if the restriction of the exponential is still surjective. I can show this works for $SL_n$ but I'm stuck on the others. This has to be written down somewhere, but I can't find any good references either.