Solve the equation: $\;$ $\log_{5-x} \left(x^2 - 2x + 65\right) = 2$
Given equation: $\;\;$ $\log_{5-x} \left(x^2 - 2x + 65\right) = 2$
i.e. $\;$ $x^2 - 2x + 65 = \left(5 - x\right)^2$
i.e. $\;$ $x^2 - 2x + 65 = 25 - 10x + x^2$
i.e. $\;$ $8x = -40$ $\implies$ $x = -5$
$\therefore \;$ The solution to the given equation is $\;\;$ $x = \left\{-5 \right\}$