Solve the equation: $\;$ $\log_4 \left\{2 \log_3 \left[1 + \log_2 \left(1 + 3 \log_3 x\right)\right] \right\} = \dfrac{1}{2}$
Given equation: $\;\;$ $\log_4 \left\{2 \log_3 \left[1 + \log_2 \left(1 + 3 \log_3 x\right)\right] \right\} = \dfrac{1}{2}$
i.e. $\;$ $2 \log_3 \left[1 + \log_2 \left(1 + 3 \log_3 x\right)\right] = 4^{\frac{1}{2}} = 2$
i.e. $\;$ $\log_3 \left[1 + \log_2 \left(1 + 3 \log_3 x\right)\right] = 1$
i.e. $\;$ $1 + \log_2 \left(1 + 3 \log_3 x\right) = 3^1 = 3$
i.e. $\;$ $\log_2 \left(1 + 3 \log_3 x\right) = 2$
i.e. $\;$ $1 + 3 \log_3 x = 2^2 = 4$
i.e. $\;$ $3 \log_3 x = 3$
i.e. $\;$ $\log_3 x = 1$
i.e. $\;$ $x = 3^1 = 3$
$\therefore \;$ The solution to the given equation is $\;\;$ $x = \left\{3 \right\}$