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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.

See The Solution Submitted by DJ    
Rating: 4.1429 (14 votes)

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Solution Solution | Comment 1 of 7
For all pairs of x, y where both are greater than or equal to e (2.71828...), the question is equivalent to taking the natural log (ln) of both sides and determining the inequality relationship there.

Taking the log of both sides, the question becomes:
which is greater pi * ln (e) --or-- e * ln(pi)?

The expression has two factors.... the first factor is a number, and the second factor is modified by taking the natural log of a number.

The unmodified factor will go up faster (as a function) than the natural log will go up, for numbers greater than or equal to e.

Therefore, to maximize the result, we would want to use the LARGER number (in this case pi) as the unmodified factor.

The unmodified factor corresponds to the exponent in the original equation.

Therefore e^pi must be greater than pi^e.

(And, in fact, it is.)
Edited on September 26, 2003, 2:24 pm
  Posted by SilverKnight on 2003-09-26 14:17:10
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