e^pi Greater Than pi^eDate: 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/ |
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