
Re: x^2 = 2^x and x^4 = 4^x
Posted:
Oct 30, 2013 10:29 PM


On Mon, 28 Oct 2013, James Ward wrote:
> The two equations: > x^2 = 2^x and x^4 = 4^x,
They're the some equation.
> both have 3 identical real solutions: > x = 2, 4, and infinite power tower of (1/sqrt(2))
x^2 = 2^x; 2.log a = x.log 2; (log x)/x = (log 2)/2
Let a = exp (log 2)/2 Define by induction a1 = a, a_(j+1) = a^aj. Let x = lim(j>oo) aj.
log x = log lim(j>oo) a_(j+1) = lim(j>oo) log a^aj . . = lim(j>oo) aj.log a = log a * lim(j>oo) aj = x.log a
(log x)/x = log a = (log 2)/2. 2.log x = x.log 2 x^2 = exp 2.log x = exp x.log 2 = 2^x
Does x = lim(j>oo) aj exist?

