In order to represent a discrete instantaneous displacement of $A$ at time $t_0$ in a differential equation I've been taught to write: $$ f'(t) = A \delta(t-t_0) $$
I was wondering how to represent a instantaneous product for a factor $A$ at the time $t_0$ in order to describe a situation like: $$ f(t_0^+) = A f(t_0^-) $$
With some naive reasoning I reached the conclusion it should be something like: $$ f'(t) = \ln (A)\delta(t-t_0)f(t) $$
How can I prove this in a almost rigorously way?
Does the representation really matter? From what I see, you deal with non-smooth ODEs. There are a bunch of ways to deal with those. I am not familiar with most of the ways, but there is a formulation in terms of complentarity that can work for that. Otherwise, if you want to solve that, I would suggest to check event capturing schemes.