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Topic: x^2 = 2^x and x^4 = 4^x
Replies: 5   Last Post: Oct 30, 2013 10:29 PM

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William Elliot

Posts: 1,706
Registered: 1/8/12
Re: x^2 = 2^x and x^4 = 4^x
Posted: Oct 30, 2013 10:29 PM
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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?





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