Question 9
a) Show that if $a,b\gt 1$ are distinct real numbers then
$$\log_a(\log_a b)\gt \log_b(\log_a b).$$
b) Let $a_1\gt a_2\gt \dots\gt a_n\gt 1$, $n\ge2$. Prove that
$$\begin{gathered}\log_{a_1}(\log_{a_1}(a_2)+\log_{a_2}(\log_{a_2}(a_3)+\dots\\+\log_{a_{n-1}}(\log_{a_{n-1}}(a_n)+\log_{a_n}(\log_{a_n}(a_1))))\gt 0.\end{gathered}$$
$$\log_a(\log_a b)\gt \log_b(\log_a b).$$
b) Let $a_1\gt a_2\gt \dots\gt a_n\gt 1$, $n\ge2$. Prove that
$$\begin{gathered}\log_{a_1}(\log_{a_1}(a_2)+\log_{a_2}(\log_{a_2}(a_3)+\dots\\+\log_{a_{n-1}}(\log_{a_{n-1}}(a_n)+\log_{a_n}(\log_{a_n}(a_1))))\gt 0.\end{gathered}$$