CAT 2017 Quantitative Ability

$\Rightarrow \frac{\log \left(\mathit{x}-3\right)}{\log 3^2}=\frac{\log \left(\mathit{x}-7\right)}{\log 3}$
$\Rightarrow \frac{\log \left(\mathit{x}-3\right)}{2\log 3}=\frac{\log \left(\mathit{x}-7\right)}{\log 3}$
$\Rightarrow \log \left(\mathit{x}-3\right)=2\log \left(\mathit{x}-7\right)$
$\Rightarrow \left(\mathit{x}-3\right)={\left(\mathit{x}-7\right)}^2$
$\Rightarrow \mathit{x}-3={\mathit{x}}^2-14\mathit{x}+49$
${\mathit{x}}^2-15\mathit{x}+52=0$
Discriminant = ${15}^2-4\times 52=225-208=17>0$
Two distinct values of x exists
However, since we have the log function involved here, we will have to verify the solutions. Log is defined only for positive terms So. if log(x - 3) and log(x - 7) are defined, x has to be greater than 7.
For the equation ${\mathit{x}}^2-15\mathit{x}+52=0$ one of the values $\frac{1}{2}\left(15-\sqrt{17}\right)$
This is less than 7. Since x cannot be less than 7 there is only one possible solution