Question

If $lo{g}_a\left(ab\right)=x$, then $lo{g}_b\left(ab\right)$ is equal to

• A. $\frac{1}{x}$
• B. $\frac{x}{1+x}$
• C. $\frac{x}{x-1}$
• D. $\frac{x}{1-x}$

Answer 1

The correct option is C $\frac{x}{x-1}$

If $y=lo{g}_b\left(ab\right)$ then $\frac{1}{x}+\frac{1}{y}=lo{g}_{ab}a+lo{g}_{ab}b$
or $\frac{1}{x}+\frac{1}{y}lo{g}_{ab}\left(ab\right)=1$
$\therefore \frac{1}{y}=1-\frac{1}{x}=\frac{x-1}{x}$ $\therefore y=\frac{x}{x-1}$.