Giải thích các bước giải:

$\log _3\left(\dfrac{3}{x}\right).\log _2x-\log _3\left(\dfrac{x^3}{\sqrt{3}}\right)=\dfrac{1}{2}\log _2\sqrt{x}$

$\to \log _3\left(\dfrac{3}{x}\right).\log _2x-\log _3\left(\dfrac{x^3}{\sqrt{3}}\right)=\dfrac{1}{2}\log _2\sqrt{x}$

$\to 2\log _2\left(x\right)\left(1-\log _3\left(x\right)\right)-2\left(\log _3\left(x^3\right)-\dfrac{1}{2}\right)=\log _2\left(\sqrt{x}\right)$

$\to 2\cdot \dfrac{\log _3\left(x\right)}{\log _3\left(2\right)}\left(1-\log _3\left(x\right)\right)-2\left(\log _3\left(x^3\right)-\dfrac{1}{2}\right)=\dfrac{\log _3\left(\sqrt{x}\right)}{\log _3\left(2\right)}$

$\to \dfrac{2\log _3\left(x\right)\left(1-\log _3\left(x\right)\right)}{\log _3\left(2\right)}-2\left(\log _3\left(x^3\right)-\dfrac{1}{2}\right)=\dfrac{\log _3\left(\sqrt{x}\right)}{\log _3\left(2\right)}$

$\to \dfrac{2\log _3\left(x\right)\left(1-\log _3\left(x\right)\right)}{\log _3\left(2\right)}-2\left(3\log _3\left(x\right)-\dfrac{1}{2}\right)=\dfrac{\log _3\left(\sqrt{x}\right)}{\log _3\left(2\right)}$

$\to \dfrac{2u\left(1-u\right)}{\log _3\left(2\right)}-2\left(3u-\dfrac{1}{2}\right)=\dfrac{\dfrac{1}{2}u}{\log _3\left(2\right)}, \log_3x=u$

$\to \dfrac{2u\left(1-u\right)}{\log _3\left(2\right)}-2\left(3u-\dfrac{1}{2}\right)=\dfrac{\dfrac{1}{2}u}{\log _3\left(2\right)}$

$\to \dfrac{2u\left(1-u\right)}{\log _3\left(2\right)}\log _3\left(2\right)-2\left(3u-\dfrac{1}{2}\right)\log _3\left(2\right)=\dfrac{\dfrac{1}{2}u}{\log _3\left(2\right)}\log _3\left(2\right)$

$\to 2u\left(1-u\right)-2\log _3\left(2\right)\left(3u-\dfrac{1}{2}\right)=\dfrac{1}{2}u$

$\to -2u^2+\left(\dfrac{3}{2}-6\log _3\left(2\right)\right)u+\log _3\left(2\right)=0$

$\to u=-\dfrac{-3+12\log _3\left(2\right)+2\sqrt{36\log _3\left(2\right)^2-10\log _3\left(2\right)+\dfrac{9}{4}}}{8},\:u=\dfrac{3-12\log _3\left(2\right)+2\sqrt{36\log _3\left(2\right)^2-10\log _3\left(2\right)+\dfrac{9}{4}}}{8}$

$\to x=\dfrac{1}{3^{\dfrac{-3+12\log _3\left(2\right)+\sqrt{144\log _3\left(2\right)^2-40\log _3\left(2\right)+9}}{8}}},\:x=3^{\dfrac{3-12\log _3\left(2\right)+2\sqrt{36\log _3\left(2\right)^2-10\log _3\left(2\right)+\dfrac{9}{4}}}{8}}$