
Re: x^2 = 2^x and x^4 = 4^x
Posted:
Oct 28, 2013 11:04 AM


On Monday, October 28, 2013 6:30:46 AM UTC7, James Ward wrote: > 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
It follows by simple algebra: if x^2 = 2^x, then (x^2)^2 = (2^x)^2, so x^4 = (2^2)^x = 4^x. In fact, for any positive integer n we have that x^2 = 2^x implies x^(2n) = (2^n)^x. It is even true for positive noninteger values of n. No need for Wolfram Alpha here.

