e^pi Greater Than pi^e

Date: 03/29/2003 at 14:37:10
From: Emma
Subject: Exponentials Proofs without Calculators

How can I show that e^pi (as in pi = 3.14...) is larger than pi^e 
without using a calculator?

It seems derivatives would not be appropriate here, because both terms 
represent constant values. Also, you can't take the ln of both (using 
an inequality) because you can't find the ln of pi without a 
calculator.

Date: 03/29/2003 at 17:15:27
From: Doctor Jerry
Subject: Re: Exponentials Proofs without Calculators

Hi Emma,

We conjecture that pi^e < e^{pi}.  This is true if and only if

   ln(pi^e) < ln(e^{pi}).

This is true if and only if

   e*ln(pi)  < pi.

This is true if and only if

(*)  e  < pi/ln(pi).

So, we may think of the function f(x) = x/ln(x).

We know that e < pi.  Find the minimum of f.  Inequality (*) will 
follow.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/