Which one is greater in each cases $\log_{2}(3)$ and $\log_{3}(4).$

Question

Evaluation of Which one is greater in each cases. $(a)\; \; \log_{2}(3)$ and $\log_{3}(11)$

$(b)\;\; \log_{3}(5)\;\;\;$ and $\;\; \log_{2}(3) \;\;\;\;\;\;(c)\;\; \log_{2}(3)\;\;\;\;$ and $\;\; \log_{3}(4)$

$\bf{My\; Solution \; (a)}$ Here $$\log_{2}(3)<\log_{2}(4) = 2$$ and $$\log_{3}(11)>\log_{3}(9) = 2$$

So we get $$\log_{2}(3)<2 < \log_{3}(11).$$

I did not understand How can I calculate which one is greater in last $2$ cases.

Help me, Thanks

Answer 1

Hint

$\log_2(3) = \frac{1}{2} \log_2(9) > \frac{1}{2}(3)$.

Do the same for the $\log_3(5)$.

The same trick works for the other pair also.