Property of the Dirac delta

Question

I'm trying to see if the following identity is true: $$ \mathrm{exp} \left(a x_0 \dfrac{\partial}{\partial x_0}\right)\delta(x-x_0) = \delta\left(x-\mathrm{e}^a x_0\right)$$ where $a$ is a constant. If it is true, how should I prove it? I tried using this well known property: $$\delta(g(x))=\sum_j \frac{\delta(x-x_j)}{|g'(x_j)|},$$ but it led to nowhere (or maybe I just got very confused with notation), and I don't see any other useful properties of the delta for this. Help appreciated!