Solve the equation: $\;$ $\dfrac{1 - 2 \left(\log x^2\right)^2}{\log x - 2 \left(\log x\right)^2} = 1$
Given equation: $\;\;$ $\dfrac{1 - 2 \left(\log x^2\right)^2}{\log x - 2 \left(\log x\right)^2} = 1$
i.e. $\;$ $1 - 2 \left(2 \log x\right)^2 = \log x - 2 \left(\log x\right)^2$
i.e. $\;$ $1 - 8 \left(\log x\right)^2 = \log x - 2 \left(\log x\right)^2$
i.e. $\;$ $6 \left(\log x\right)^2 + \log x - 1 = 0$
i.e. $\;$ $6 \left(\log x\right)^2 + 3 \log x - 2 \log x - 1 = 0$
i.e. $\;$ $3 \log x \left(2 \log x + 1\right) - 1 \left(2 \log x + 1\right) = 0$
i.e. $\;$ $\left(3 \log x - 1\right) \left(2 \log x + 1\right) = 0$
i.e. $\;$ $\log x = \dfrac{1}{3}$ $\;$ or $\;$ $\log x = \dfrac{-1}{2}$
$\implies$ $x = 10^{\frac{1}{3}} = \sqrt[3]{10}$ $\;$ or $\;$ $x = 10^{\frac{-1}{2}} = \dfrac{1}{\sqrt{10}}$
$\therefore \;$ The solution to the given equation is $\;\;$ $x = \left\{\dfrac{1}{\sqrt{10}}, \; \sqrt[3]{10} \right\}$