Comment by SassyBird

It doesn’t matter which definition you pick. I e.g. had it defined axiomatically when I studied, i.e. we were given a list of properties which identify exp unambiguously and then we were given a proof of its existence. The fact that those properties were part of the definition doesn’t take away from their profoundness. The function could still just not exist. Same thing with defining by formulas. The map is not the territory.