Logarithms equality (log_2(n))^(log_2(n))=n^log_2(log_2(n))

Question

How can I prove (log_2(n))^(log_2(n))=n^log_2(log_2(n)). I tried already everything, but I don't know how to prove it mathematically, if I put these two in desmos I get the same function/

Answer 1

To prove a result of the form $\color{red}{a}^\color{blue}{b}=\color{limegreen}{c}^\color{hotpink}{d}$ take logarithms of both sides, say base-$2$ logarithms. In this problem the left-hand side becomes $\color{blue}{\log_2n}\color{red}{\log_2\log_2n}$, while the right-hand side becomes $\color{hotpink}{\log_2\log_2n}\color{limegreen}{\log_2n}$.