What function does $P(2^{2^{2^s}},2^{2^{2^{-s}}})$ trace out?

Question

This is related: Do these points trace out a function? $ P(2^{2^s},2^{2^{-s}}) $

What function does $P(2^{2^{2^s}},2^{2^{2^{-s}}})$ trace out?

I tried going through the answer that was given in the previous post but could not figure it out for this extension of the problem. I don't understand how to find the function.

Answer 1

Let's split this into a few steps:

• We can get from $x:=\color{red}{2^s}$ to $y:=\color{red}{2^{-s}}$ with $\color{red}{y=1/x}$.
• We can get from $x:=\color{orange}{2}^{\color{red}{2^s}}$ to $y:=\color{orange}{2}^{\color{red}{2^{-s}}}$ with $\color{red}{\log_2y=1/\log_2x}$, i.e. $\color{orange}{y=2^{1/\log_2x}}$.
• We can get from $x:=\color{limegreen}{2}^{\color{orange}{2}^{\color{red}{2^s}}}$ to $y:=\color{limegreen}{2}^{\color{orange}{2}^{\color{red}{2^{-s}}}}$ with $\color{orange}{\log_2y=2^{1/\log_2\log_2x}}$, i.e. $\color{limegreen}{y=2^{2^{1/\log_2\log_2x}}}$.