An algebra problem by أحمد الحلاق

$\begin{aligned} Q & = \frac {\log_{10}20^2\cdot \log_{20}30^2\cdot \log_{30}40^2\cdots \log_{990}1000^2}{\log_{10}11^2\cdot \log_{11}12^2\cdot \log_{12}13^2\cdots \log_{99}100^2} \\ & = \frac {\frac {2 \cancel{\log 20}}{\log 10}\cdot \frac {2 \cancel{\log 30}}{\cancel{\log 20}}\cdot \frac {2\cancel{\log 40}}{\cancel{\log 30}}\cdots \frac {2\log 1000}{\cancel{\log 990}}}{\frac {2 \cancel{\log 11}}{\log 10}\cdot \frac {2 \cancel{\log 12}}{\cancel{\log 11}}\cdot \frac {2\cancel{\log 13}}{\cancel{\log 12}}\cdots \frac {2\log 100}{\cancel{\log 99}}} \\ & = \frac {2^{99}\frac {\log 1000}{\log 10}}{2^{90}\frac {\log 100}{\log 10}} = \frac {2^{99}\cdot 3}{2^{90}\cdot 2} = \boxed{768} \end{aligned}$