Compare $\log_2 3$ and $\log_3 5$ without using a calculator.
I am not very good at math please explain it clearly
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Sign up to join this communityCompare $\log_2 3$ and $\log_3 5$ without using a calculator.
I am not very good at math please explain it clearly
Note: $$\log_2 3=\frac14 \log_2 81>\frac14\log_2 64=\frac64=\frac32,$$ $$\log_3 5=\frac14 \log_3 625<\frac14\log_3 729=\frac64=\frac32.$$
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!
$$\sqrt{3}\approx 1.73,\sqrt{2}\approx 1.42\\2^{1.5}=2\sqrt{2}< 2.84,\ 3^{1.5}=3\sqrt{3}>5.19\\\log_2 3>\log_22.84>\log_22\sqrt2= 1.5=\log_33\sqrt{3}> \log_35.19>\log_3 5$$