Explanation

(d)
We have, $\frac{\log{5 + log(x^{2} + 1)}}{\log{(x - 2)}} = 2$
$\Rightarrow \log\left\{ 5\left( x^{2} + 1 \right) \right\} = {\log(x - 2)}^{2}$
$\Rightarrow 5\left( x^{2} + 1 \right) = (x - 2)^{2}$
$\Rightarrow 4x^{2} + 4x + 1 = 0$
$\Rightarrow x = - \frac{1}{2}$
But for $x = - \frac{1}{2},\log{(x - 2)}$ is not meaningful.
$\therefore$ It has no root.