If $a$ and $b$ is $> 1$, and $p=\frac{\log_{b}(\log_{b}{a})}{\log_ba}$ what is $a^p$?

Question

If $a$ and $b$ is $> 1$, and $p=\frac{\log_{b}(\log_{b}{a})}{\log_ba}$ what is $a^p$?

I wasn't even sure on how to start (don't vote down simply because I don't show work I do not know on how to start the problem

Answer 1

Using the change of basis formula, namely $\log_a X=\dfrac{\log_b X}{\log_b a}$, we get $$p=\log_a\left(\log_ba\right)$$ So $a^p=\log_ba$.

Answer 2

Put $c=\log_b b$ i.e. $b^c=a$: $$\begin{align} p&=\frac {\log_b(\log_b a)}{\log_b a} =\frac {\log_b c}c\\ cp&=\log_b c\\ c&=b^{cp} =(b^c)^p =a^p\\ a^p&=\color{red}{\log_b a}\end{align}$$