Question

Evaluate $\frac{{\log }_{40}1000}{{\log }_{40}100}$

Answer 1

Let $a=\frac{{\log }_{40}1000}{{\log }_{40}100}$

We know that,

$1000=(10{)}^3$ $100=(10{)}^2$

$\therefore a=\frac{{\log }_{40}{10}^3}{{\log }_{40}{10}^2}$

$=\frac{3{\log }_{40}10}{2{\log }_{40}10}\dots (\because n{\log }_{a}b={\log }_a{b}^n)$

$a=\frac{3}{2}$

$\therefore \frac{{\log }_{40}1000}{{\log }_{40}100}=\frac{3}{2}$