TITLE: Solving equation $\log_y(\log_y(x))= \log_n(x)$ for $n$
QUESTION [6 upvotes]: I'm just wondering, if I log a constant twice with the same base $y$,
$$\log_y(\log_y(x))= \log_n(x)$$
Can it be equivalent to logging the same constant with base $n$? If yes, what is variable $n$ equivalent to?

REPLY [5 votes]: $$\log_y(\log_y(x))= \log_n(x)$$ $$\implies \log_y(\log_y(x))= \frac{\log_e(x)}{\log_e(n)}$$ $$\implies \log_e(n)=\frac{\log_e(x)}{\log_y(\log_y(x))}$$ $$\implies n=e^{\frac{\log_e(x)}{\log_y(\log_y(x))}}$$
Thus, for given $x,y$, if $e^{\frac{\log_e(x)}{\log_y(\log_y(x))}}$ is defined, then that is the value for $n$.