Solve the equation: $\;$ $4^{\frac{1}{x} - 2} = \dfrac{\log \sqrt{10}}{2}$
Given equation: $\;\;$ $4^{\frac{1}{x} - 2} = \dfrac{\log \sqrt{10}}{2}$
i.e. $\;$ $4^{\frac{1}{x} - 2} = \dfrac{1}{2} \times \log_{10} {10}^{\frac{1}{2}}$
i.e. $\;$ $4^{\frac{1}{x} - 2} = \dfrac{1}{2} \times \dfrac{1}{2} \times \log_{10} 10$
i.e. $\;$ $4^{\frac{1}{x} - 2} = \dfrac{1}{4} \times 1$
i.e. $\;$ $4^{\frac{1}{x} - 2} \times 4^1 = 1$
i.e. $\;$ $4^{\frac{1}{x} - 1} = 4^0$
$\implies$ $\dfrac{1}{x} - 1 = 0$
i.e. $\;$ $\dfrac{1}{x} = 1$ $\implies$ $x = 1$
$\therefore \;$ The solution to the given equation is $\;\;$ $x = \left\{1 \right\}$