Solve for x:

$\dfrac{\text{log 8}}{\text{log 2}} \times \dfrac{\text{log 3}}{\text{log}\sqrt{3}} = 2\text{log} x$

Given,

$\Rightarrow \dfrac{\text{log 8}}{\text{log 2}} \times \dfrac{\text{log 3}}{log\sqrt{3}} = 2log x \\[1em] \Rightarrow \dfrac{\text{log 2}^3}{\text{log 2}} \times \dfrac{\text{log 3}}{\text{log} 3^{\dfrac{1}{2}}} = 2log x \\[1em] \Rightarrow \dfrac{\text{3log 2}}{\text{log 2}} \times \dfrac{\text{log 3}}{\dfrac{1}{2}\text{log 3}} = \text{2 log x} \\[1em] \Rightarrow 3 \times 2 = 2\text{log } x \\[1em] \Rightarrow \text{log } x = \dfrac{6}{2} \\[1em] \Rightarrow \text{log } x = 3 \\[1em] \Rightarrow x = 10^3 = 1000.$

Hence, x = 1000.