\[ e^\pi> \pi^e.\]

How would we go about convincing someone that this amazing inequality is true?

There are several ways to get a feel for this inequality, here are some suggestions:

  1. what do the functions \(e^x\) and \(x^e\) look like?
  2. how could we approximate \(e\) and \(\pi\)?
  3. do we know any functions that change a question about indices into a question about something else?

How do we define \(e^\pi\) and \(\pi^e\) given that both numbers are transcendental?