From: utzoo!decvax!decwrl!sun!megatest!fortune!hpda!hplabs!hao!seismo!harpo!floyd!whuxlb!eisx!npoiv!houxm!houxz!halle1
Newsgroups: net.math
Title: Infinite exponentiation
Article-I.D.: houxz.301
Posted: Tue May 10 17:13:38 1983
Received: Tue May 17 00:54:42 1983

x**x**x**...=n has a solution for all n >= 0.  (n=0 is an asymptotic
solution, i.e. lim n->0)
>From the earlier article, x**x**x**x...=x**(x**x**...)=n  ==> x**n=n.

Taking logs:  n ln(x)= ln(n)  or  ln(x)= ln(n) /n
	Therefore: x=exp( ln(n) /n) = (exp( ln(n))**(1/n)
	=>    x=n**(1/n)

                1/n
	   x = n      is defined for all n>o, equals 0 at n=0, and is
undefined (in the real plane) for n<0.  (Special cases excepted.)

The maximum value of x occurs at n=e (2.71828182859....)  x=1.4447  (rounded).