<p>Which is bigger: e^pi or pi^e? I know the answer is e^pi, but I need to do a formal proof. Any ideas?</p>
<p>I would expand out pi and e.</p>
<p>There probably is a quicker way, but try this:</p>
<p>pi^e is the same as e^(e ln pi); so you just need to show
e^pi > e^(e ln pi), or (looking at the exponents)
show that pi > e ln pi .</p>
<p>If you plot f(x) = x and g(x) = e ln(x), you will find</p>
<p>a) for 0 < x <= 1, ln(x) is < 0, so x will always > e ln(x) here
b) for large values of x, x >> e ln(x)
c) if you can show that f(x) - g(x) is minimized at some x=b, and that
f(b) - g(b) is > 0, then f(x) is always > g(x). In particular, f(pi) > e ln(pi), so QED .</p>
<p>This can be shown, since f(x) - g(x) = x - e ln(x), and so
f'(x) - g'(x) = 1 - e(1/x) ; this is equal to 0 when x=e, and > 0 when x > e
f(e) - g(e) = e - e ln(e) = 0.</p>
<p>So, x - e ln(x) = 0 when x=e, and is > 0 when x > e.</p>
<p>This would be a better place for your question:
<a href="http://physicsforums.com/forumdisplay.php?f=15%5B/url%5D">http://physicsforums.com/forumdisplay.php?f=15</a></p>
<p>Please don't take it personally - just can't help making a pun:
one neo can ask such questions that forty-dudes won't help.
Since there is a hint of urgency in your request,
here goes the answer.</p>
<p>Let's examine the function f(x) = ln(x) / x.
f'(x) = ((1/x) (x) - ln(x) (1)) / (x^2) =
(1 - ln x) / x^2</p>
<p>For x>e f'(x)<0 and f(x) is decreasing:
If x1 > e then f(x1) < f(e).</p>
<p>Since pi > e,
f(pi) < f(e).</p>
<p>ln(pi) / pi < ln(e) / e
e ln(pi) < pi ln(e)
ln(pi^e) < ln(e^pi)
pi^e < e^pi.
QED</p>
<p>Gcf101:
...what does it tell you about two guys who use QED as a reflex when solving a non-trivial problem? :)</p>
<p>Optimizerdad - i hope you'll believe me - I was trying to type this solution from my memory, and when I hit "submit", there was you already! :o
Chances are high that theoneo and gcf101 are two different guys (or girls, or a girl and a guy, and many more possibilities...), so there was no trap-setting involved. :)</p>
<p>Instead of QED I should've used QEF ("which was to be done"):
are not we all here driven by the sense of duty? ;)</p>