
x^2 = 2^x and x^4 = 4^x
Posted:
Oct 28, 2013 9:30 AM


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

