We have, \( \Large \frac{log5 + log \left(x^{2}+1\right) }{log \left(x-2\right) } = 2 \) => \( \Large log\{ 5 \left(x^{2}+1\right) \} = log \left(x-2\right)^{2} \) => \( \Large 5 \left(x^{2}+1\right)= \left(x-2\right)^{2}=4x^{2}+4x+1=0 \) => \( \Large x = -\frac{1}{2} \) But for \( \Large x = -\frac{1}{2},\ log \left(x-z\right) \) is not meaningful, it has no root.