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e and pi (Posted on 2003-09-26) Difficulty: 4 of 5
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

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionAn Alternative MethodologyK Sengupta2007-05-25 11:36:44
re: Comment on official solutionThoughtProvoker2004-06-13 15:02:09
Some ThoughtsComment on official solutionNick Hobson2004-06-13 14:05:45
SolutionBy power seriesNick Hobson2003-10-09 17:08:14
re: SolutionSilverKnight2003-09-26 14:32:23
SolutionBrian Smith2003-09-26 14:30:43
SolutionSolutionSilverKnight2003-09-26 14:17:10
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