\[\begin{array}{l}
 {\log ^2}x + 2\log x\log y + {\log ^2}y = 10 + 2\log x\log y \\ 
 {\left( {\log x + \log y} \right)^2} = 10 + 2\log x\log \frac{{100}}{x} \\ 
 {\log ^2}xy = 10 + 2\log x\left( {\log {{10}^2} - \log x} \right) \\ 
 4 = 10 + 4\log x - 2{\log ^2}x \\ 
 2{\log ^2}x - 4\log x - 6 = 0/:2 \\ 
 \log x = t \\ 
 {t^2} - 2t - 3 = 0 \\ 
 \end{array}\]