Question

If ${\log }_a\left(ab\right)=x,$ then evaluate ${\log }_b\left(ab\right)$ in terms of $x$.
if $x=2$ then its value is ?

Answer 1

If $y={\log }_b\left(ab\right),$ then
$\frac{1}{x}+\frac{1}{y}={\log }_{ab}a+{\log }_{ab}b$
or $\frac{1}{x}+\frac{1}{y}={\log }_{ab}\left(ab\right)=1$
$\therefore \frac{1}{y}=1-\frac{1}{x}=\frac{x-1}{x}\therefore y=\frac{x}{x-1}=2$
Ans: 2