WMC 2018 Pursuit - Pack 1, A2

$\begin{aligned} {\color{#3D99F6}\frac 12 \log_{4x} 16} - 3\log_{16x} 4 & = 0 & \small \color{#3D99F6} \text{By }a \log b = \log b^a \\ \log_{4x} 4 - 3 \log_{16x} 4 & = 0 & \small \color{#3D99F6} \text{Change to a common base} \\ \frac {\log 4}{\log (4x)} - \frac {3 \log 4}{\log (16x)} & = 0 & \small \color{#3D99F6} \text{Divide both sides by }\log 4 \text{ and rearrange} \end{aligned}$

$\begin{aligned} \implies \frac 1{\log (4x)} & = \frac 3{\log (16x)} \\ \log (16x) & = 3 \log (4x) \\ \log (16x) & = \log (64x^3) \\ \implies 16x & = 64 x^3 & \small \color{#3D99F6} \text{Since }x \ne 0 \\ x^2 & = \frac 14 \\ x & = \frac 12 = \boxed {0.5} & \small \color{#3D99F6} \text{Since }x > 0 \end{aligned}$