Solve the equation: $\;$ $2 \left(\log_x \sqrt{5}\right)^2 - 3 \log_x \sqrt{5} + 1 = 0$
Given equation: $\;\;$ $2 \left(\log_x \sqrt{5}\right)^2 - 3 \log_x \sqrt{5} + 1 = 0$
i.e. $\;$ $2 \left(\log_x \sqrt{5}\right)^2 - 2 \log_x \sqrt{5} - \log_x \sqrt{5} + 1 = 0$
i.e. $\;$ $2 \log_x \sqrt{5} \left(\log_x \sqrt{5} - 1\right) - 1 \left(\log_x \sqrt{5} - 1\right) = 0$
i.e. $\;$ $\left(2 \log_x \sqrt{5} - 1\right) \left(\log_x \sqrt{5} - 1\right) = 0$
i.e. $\;$ $\log_x \sqrt{5} = \dfrac{1}{2}$, $\;$ or $\;$ $\log_x \sqrt{5} = 1$
i.e. $\;$ $\sqrt{5} = x^{\frac{1}{2}} = \sqrt{x}$ $\;$ or $\;$ $\sqrt{5} = x^1= x$
$\implies$ $x = 5$ $\;$ or $\;$ $x = \sqrt{5}$
$\therefore \;$ The solution to the given equation is $\;\;$ $x = \left\{\sqrt{5}, \; 5 \right\}$