Question

\[\begin{vmatrix}\log_3 512 & \log_4 3 \\ \log_3 8 & \log_4 9\end{vmatrix} \times \begin{vmatrix}\log_2 3 & \log_8 3 \\ \log_3 4 & \log_3 4\end{vmatrix}\]

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Answer 1

\[\begin{vmatrix}\log_3 512 & \log_4 3 \\ \log_3 8 & \log_4 9\end{vmatrix} \times \begin{vmatrix}\log_2 3 & \log_8 3 \\ \log_3 4 & \log_3 4\end{vmatrix}\]
\[ = \begin{vmatrix} \log_3 2^9 & \log_{2^2} 3 \\ \log_3 2^3 & \log_{2^2} 3^3 \end{vmatrix} \times \begin{vmatrix} \log_2 3 & \log_{2^3} 3\\ \log_3 2^2 & \log_3 2^2 \end{vmatrix}\]
\[ = \begin{vmatrix} 9 \log_3 2 & \frac{1}{2} \log_2 3 \\ 3 \log_3 2 & \frac{1}{2} \times 2 \log_2 3 \end{vmatrix} \times \begin{vmatrix} \log_2 3 & \frac{1}{3} \log_2 3\\ 2 \log_3 2 & 2 \log_3 2 \end{vmatrix} \left[ \because \log {}_{b^m} a^n = \frac{n}{m} \log_b a \right]\]
\[ = \left( \left( 9 \log_3 2 \times \log_2 3 \right) - \left( 3 \log_3 2 \times \frac{1}{2} \log_2 3 \right) \right) \times \left( \left( \log_2 3 \times 2 \log_3 2 \right) - \left( \frac{1}{3} \log_2 3 \times 2 \log_3 2 \right) \right) \left[ \because \log_m n \times \log_n m = 1 \right]\]
\[ = \left( 9 - \frac{3}{2} \right) \times \left( 2 - \frac{2}{3} \right)\]
\[ = \frac{15}{2} \times \frac{4}{3} = 10\]