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