Question

Solve the following equation:
${\log }^2\left(100x\right)+{\log }^2\left(10x\right)=14+\log \frac{1}{x}$

Answer 1

${\log }^2\left(100x\right)+{\log }^2\left(10x\right)=14+\log \frac{1}{x}\Rightarrow {(2+\log x)}^2+{(1+\log x)}^2=14-\log x\Rightarrow 2{\left(\log x\right)}^2+6\log x+5=14-\log x\Rightarrow 2({\log x)}^2+6\log x+5=14-\log x\Rightarrow 2({\log x)}^2+71\log x-9=0\Rightarrow 2(\log x+9)(\log x-1)=0\Rightarrow \log x=\frac{-9}{2},1\Rightarrow x={10}^{\frac{-9}{2}},10$