Math CentralQuandaries & Queries


Question from natalie, a student:

Without using your calculator, prove which is bigger: e^pi or pi^e

we talked about ad absurdum in class, so I'm assuming that is how I approach this question, but since neither of them have a variable( like the examples we actually solved in class), I'm not quite sure how to solve it....


Look at the last paragraph on
and see if that helps. If you need more assistance write back.


Natalie wrote back


I'm still confused. How does creating a (lnx)/x help us get to the conclusion? I'm not even sure why it is a graph of (lnx)/x when it is about e^pi and pi^e??

thank you,


If f(x) = ln(x)/x and the maximum value of f(x) occurs at x = e the f(e) > f(π). Thus

ln(e)/e > ln(π)/π.

But ln(e) = 1 so

1/e > ln(π)/π.

Can you manipulate this inequality to show that eπ > πe? If so then all that remains is to show that the maximum value of f(x) occurs at x = e. I used calculus.


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