TITLE: How can I compare $\log_2 3$ and $\log_3 5$ without using a calculator
QUESTION [5 upvotes]: Compare $\log_2 3$ and $\log_3 5$ without using a calculator.

I am not very good at math please explain it clearly

REPLY [9 votes]: Note:
$$\log_2 3=\frac14 \log_2 81>\frac14\log_2 64=\frac64=\frac32,$$
$$\log_3 5=\frac14 \log_3 625<\frac14\log_3 729=\frac64=\frac32.$$

REPLY [3 votes]: We'll prove that $$\log_35<\log_23$$ or
$$5<3^{\log_23}$$ or
$$25<3^{\log_29},$$
which is true because
$$3^{\log_29}>3^{\log_28}=27>25.$$
Done!