little-oh/big-oh rules for function composition

Question

I'm confused on the general rules for little-oh and big-oh when doing function composition. For example, in Zorich analysis it is proven that $e^x -1 = x+o(x)$ as $x \to 0$. Later zorich uses this fact to directly say that $e^{h\ln a}-1 = h \ln a+o(h \ln a).$ While directly substituting $x$ with $h \ln a$ seems intuitive, I think that there is implicitly a rule of little-oh for function compositions used here in order to justify this step. So what is the rule of function composition for little-oh? I'm not asking for the proof that $e^{h\ln a}-1 = h \ln a+o(h \ln a)$, but a general rule of function composition for little-oh so that I can substitute "$x$" with another expression in freely under appropriate assumptions