Question

Solve:$\log \left(\log x\right)+\log (\log x^3-2)=0$

Answer 1

$\log \left(\log x\right)+\log (\log x^3-2)=0$

$\Rightarrow \log \left(\log x\right)(\log x^3-2)=0$

$\Rightarrow \left(\log x\right)(\log x^3-2)={10}^0=1$

$\Rightarrow \log x(3\log x-2)=1$

$\Rightarrow 3{\left(\log x\right)}^2-2\log x-1=0$

$\Rightarrow 3{\left(\log x\right)}^2-3\log x+\log x-1=0$

$\Rightarrow 3\log x(\log x-1)+1(\log x-1)=0$

$\Rightarrow (\log x-1)(3\log x+1)=0$

$\Rightarrow \log x=1,\frac{-1}{3}$

$\therefore x={10}^1,{10}^{\frac{-1}{3}}$

$\therefore x=10,\frac{1}{\sqrt[3]{310}}$