Bound on (matrix spectral norm) $\|A-B\|$ from bound on $\|e^A-e^B\|$

Question

Let $A,B$ be orthogonal matrices satisfying $\|A\|, \|B\| \geq c$. Now provided with $ \| e^A-e^B\|\leq \varepsilon e^{c} $, how do I show that $\| A-B \| \leq \varepsilon$?

My intuition comes from the fact that this holds true when $A,B$ are scalars. I thought I could maybe use the idea from the scalar case on the eigenvalues of $A,B$ or something like that...