If ln x is defined via an integral and e defined from ln ...
If is defined via an integral and defined from , how would you prove that is the inverse of ?
This is a somewhat technically specific question about the relationship between and given one possible definition of
Suppose that you define as
We can use the connection between the integral definition of and the harmonic series to show that grows unboundedly. This function is monotonically increasing, so it should have an inverse mapping from its codomain () to its domain (). Let's call that function
Since grows unboundedly and is zero when is , we can define as the unique value such that
So here's my question: given these starting assumptions, how would you prove that (or, equivalently, that ? I'm having a lot of trouble even seeing how you'd get started proving these facts, since basically every calculus fact I know about and presumes this result to be true.