Question

Prove that $lo{g}_{10}2$ lies between $\frac{1}{4}$ and $\frac{1}{3}$

Answer 1

$lo{g}_{10}2>\frac{1}{4}$ if 2 > 10${}^{1/4}$
or $2^4>10$ which is true
Again $lo{g}_{10}2>\frac{1}{3}$ if 2 > 10${}^{1/3}$ , i.e., if $2^3$ <10,
Which is again true
Hence $\frac{1}{4}<lo{g}_{10}2<\frac{1}{3}$