Explanation

(b)
$\ \left| \begin{matrix}
\log_{3}512 & \log_{4}3 \\
\log_{3}8 & \log_{4}9
\end{matrix} \right| \times \left| \begin{matrix}
\log_{2}3 & \log_{8}3 \\
\log_{3}4 & \log_{3}4
\end{matrix} \right|$
$= (\log_{3}512 \times \log_{4}9 - \log_{4}{3\ \log_{3}8)} \times (\log_{2}3 \times \log_{3}4 - \log_{8}3 \times \log_{3}4)$
$= \left( \ \frac{\log 512}{\log 3} \times \frac{\log 9}{\log 4} - \frac{\log 3}{\log 4} \times \frac{\log 8}{\log 3} \right) \times \left( \ \frac{\log 3}{\log 2} \times \frac{\log 4}{\log 3} \times \frac{\log 3}{\log 8} \times \frac{\log 4}{\log 3} \right)$
$= \left( \ \frac{\log 2^{9}}{\log 3} \times \frac{\log 3^{2}}{\log 2^{2}} \times \frac{\log 2^{3}}{{\log 2}^{2}} \right) \times \left( \ \frac{{\log 2}^{2}}{\log 2} - \frac{{\log 2}^{2}}{\log 2^{3}} \right)$
$= \left( \ \frac{9 \times 2}{2} - \frac{3}{2} \right) \times \left( \ 2 - \frac{2}{3} \right) = \frac{15}{2} \times \frac{4}{3} = 10$