Date: Oct 28, 2013 9:30 AM
Author: jimward2@gmail.com
Subject: x^2 = 2^x and x^4 = 4^x
I didn't see this curious fact mentioned in the wiki article (perhaps it is well known) http://en.wikipedia.org/wiki/Tetration

The two equations:

x^2 = 2^x and x^4 = 4^x,

both have 3 identical real solutions:

x = 2, 4, and -infinite power tower of (1/sqrt(2))

You can check the last in Wolfram Alpha using:

x = 1/sqrt(2),

y = -ProductLog((-log(x)))/(-log(x)),

z = 2^y - y^2

and

x = 1/sqrt(2),

y = -ProductLog((-log(x)))/(-log(x)),

z = 4^y - y^4