Question

Find the number of solution(s) of $\frac{1-2{\left(\log x^2\right)}^2}{\log x-2{\left(\log x\right)}^2}=1$.

Answer 1

The given equation can be rewritten in the form $\frac{1-2{\left(\log x^2\right)}^2}{\log x-2{\left(\log x\right)}^2}=1$
$\Rightarrow \frac{1-8{\left(\log x\right)}^2}{\log x-2{\left(\log x\right)}^2}-1=0$
Let $\log x=t$, then $\frac{1-{8t}^2}{t-{2t}^2}-1=0$
$\Rightarrow \frac{1-{8t}^2-t+{2t}^2}{t-{2t}^2}=0\Rightarrow \frac{1-t-{6t}^2}{(t-{2t}^2)}=0\frac{(1+2t)(1-3t)}{t(1-2t)}=0$
$\Rightarrow \{\begin{array}{c}t=-\frac{1}{2}\\ t=\frac{1}{3}\end{array}$
$\Rightarrow \{\begin{array}{c}\log x=-\frac{1}{2}\\ \log x=\frac{1}{3}\end{array}\Rightarrow \{\begin{array}{c}{x}_1={10}^{-\frac{1}{2}}\\ x_2={10}^{\frac{1}{3}}\end{array}$
Hence, $x_1=\frac{1}{\sqrt{10}}$ and $x_2=\sqrt[3]{310}$ are the roots of the original equation.