Solving $\sqrt{x}^{\sqrt{x}^{\sqrt{x}}} = 2^{512}$.

Question

I can't solve this question I have tried but I can't find any other websites that help. Thanks

$$\sqrt{x}^{\sqrt{x}^{\sqrt{x}}} = 2^{512}$$

This is not infinite exponent just 2 times

Answer 1

Hint: First define $y=\sqrt x$ to clean things up. Then if you are in the integers note that $y$ must be a power of $2$ so let $y=2^z$. Plug that in and use the laws of exponents. Taking the $\log_2$ of the equation will also clean things up a bit.

If you are in the reals you have $$y^{y^y}=2^{512}\\ y^y\log_2y=512\\y\log_2 y +\log_2(\log_2 y)=9$$ and we can do fixed point iteration with $$y=\frac {9-\log_2(\log_2 y)}{\log_2 y}$$ I find $y=4$ is a solution, so $x=16$