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 e and pi (Posted on 2003-09-26)
Without finding the numerical values, show which is greater, e^π or π^e.

 Submitted by DJ Rating: 4.1429 (14 votes) Solution: (Hide) eπ > πe. The underlying idea here is that ab > ba whenever b > a, given that a and b are both greater than 1. If you look at the graphs of y=xa and y=ax, they will cross, of course, at x=a. For higher values of x (again, given that a>1), the graph of ax rises faster than xa. Thus, when x>a, ax > xa. 1 < e < π, so eπ > πe. Another way to prove this directly follows: π > e, so ln(π) > 1 e(1-1) = 1 e(x-1) > x (for x>1) e[ln(π)-1] > ln(π) (eln(π))/e > ln(π) eln(π) > e ln(π) π > e ln(π) π > ln(πe) eπ > eln(πe) eπ > πe

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 Subject Author Date An Alternative Methodology K Sengupta 2007-05-25 11:36:44 re: Comment on official solution ThoughtProvoker 2004-06-13 15:02:09 Comment on official solution Nick Hobson 2004-06-13 14:05:45 By power series Nick Hobson 2003-10-09 17:08:14 re: Solution SilverKnight 2003-09-26 14:32:23 Solution Brian Smith 2003-09-26 14:30:43 Solution SilverKnight 2003-09-26 14:17:10

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