> This has nothing to do with e and is satified by 2^x or any a^x, so this wouldn't work for introducing e in particular.

You need to impose f'(0) = 1. (If you want to be really technical, also at least some regularity condition, I'll be honest, I don't remember what's the minimal one, let's say continuity)

> Same for this

I did say up to a constant.