If _a ( ab ) = x ,\, then _b ( ab ) is -

Answer: Option D

$$\begin{array}{rl}& \text{lo}{\text{g}}_a\left(ab\right)=x\\ & \Rightarrow \frac{\log ab}{\log a}=x\\ & \Rightarrow \frac{\log a+\log b}{\log a}=x\\ & \Rightarrow 1+\frac{\log b}{\log a}=x\\ & \Rightarrow \frac{\log b}{\log a}=x-1\\ & \Rightarrow \frac{\log a}{\log b}=\frac{1}{x-1}\\ & \Rightarrow 1+\frac{\log a}{\log b}=1+\frac{1}{x-1}\\ & \Rightarrow \frac{\log b}{\log b}+\frac{\log a}{\log b}=\frac{x}{x-1}\\ & \Rightarrow \frac{\log b+\log a}{\log b}=\frac{x}{x-1}\\ & \Rightarrow \frac{\text{log}\left(ab\right)}{\log b}=\frac{x}{x-1}\\ & \Rightarrow \text{lo}{\text{g}}_b\left(ab\right)=\frac{x}{x-1}\end{array}$$