Which is bigger among (i) $\log_2 3$ and $\log _3 5$ (ii) $\log_2 3$ and $\log _3 11$.
Which is bigger among (i) $\log_2 3$ and $\log _3 5$ (ii) $\log_2 3$ and $\log _3 11$.
Trial: I use AM-GM inequality and get $$\dfrac{\log_e 2+\log_e 5}{2}>\sqrt{\log_e 2\cdot \log_e 5} \\ \implies \dfrac{\log _e10}{2}>\sqrt{\log_e 2\cdot \log_e 5} $$ Then I can not proceed. Please help.
(i) Since $5^2=25\lt27=3^3$ and $3^2=9\gt8=2^3$, $2\log_35\lt3\lt2\log_23$ hence $\log_35\lt\log_23$.
(ii) Since $3\lt4=2^2$ and $11\gt9=3^2$, $\log_23\lt2\lt\log_311$.
For the (ii) $\log_311>\log_39=\log_3(3^2)=2$ but $\log_23<\log_24=\log_2(2^2)=2$