Prove that: e^pi > pi^e, without using a calculator

[Parth Dave]

How can you prove that:
e^pi > pi^e,
without using a calculator. (all you know is the values for e and pi.)

 

[mathwonk]

well that says that e^pi > e^(e ln(pi)), so since e^x is increasing you just have to show that pi > e ln(pi). so we just need to show that pi/ln(pi) > e.

but x/ln(x) is increasing for x >=e, and e/ln(e) = e, and pi > e, so pi/ln(pi) > e/ln(e) = e. qed.

thats cute, I'm going to try that on my calculus class.