The value of \(\left|\begin{array}{cc}\log _{3} 1024 & \log _{8} 3 \\ \log _{3} 8 & \log _{4} 9\end{array}\right| \times\left|\begin{array}{ll}\log _{2} 3 & \log _{4} 3 \\ \log _{3} 4 & \log _{3} 4\end{array}\right|=\ldots\) (a) 6 (b) 9 (c) 10 (d) 12

We start by simplifying the elements in both matrices using logarithm properties, such as change of base and exponentiation. We can change the base using the formula: \(\log_b a = \frac{\log_c a}{\log_c b} \), where c is a common base we want to use, in this case, 2. The elements of the given matrices can be simplified as follows: Matrix 1: - \(\log_3 1024 = \log_{2^2} 2^{10} = \frac{10 \cdot \log_2 2}{2 \cdot \log_2 3} = \frac{10}{2} = 5\) - \(\log_8 3 = \log_{2^3} 3 = \frac{\log_2 3}{3 \cdot \log_2 2} = \frac{1}{3}\) - \(\log_3 8 = \log_{3} 2^3 = 3 \cdot \frac{\log_2 2}{\log_2 3} = \frac{3}{\log_2 3}\) - \(\log_4 9 = \log_{2^2} 3^2 = \frac{2 \cdot \log_2 3}{2 \cdot \log_2 2} = \log_2 3\) Matrix 2: - \(\log_2 3\): keep as-is - \(\log_4 3 = \log_{2^2} 3 = \frac{\log_2 3}{2 \cdot \log_2 2} = \frac{1}{2} \cdot \log_2 3\) - \(\log_3 4 = \log_{3} 2^2 = 2 \cdot \frac{\log_2 2}{\log_2 3} = \frac{2}{\log_2 3}\) - \(\log_3 4 = \frac{2}{\log_2 3}\): repeated element

Now, we calculate the determinants for each simplified matrix: Matrix 1 determinant: \(D_1 = |5 \hphantom{space} 1/3| = 5 \cdot \log_2 3 - 1/3 \cdot 3/\log_2 3 = 5 \log_2 3 - 1\) \(|\hphantom{spacespace} \frac{3}{\log_2 3} \hphantom{space} \log_2 3|\) Matrix 2 determinant: \(D_2 = |\log_2 3 \hphantom{space} 1/2 \cdot \log_2 3| = \log_2 3 \cdot \log_2 3 - 1/2 \cdot \log_2 3 \cdot 2/\log_2 3 = \log_2 3 (\log_2 3 - 1)\) \(|\frac{2}{\log_2 3} \hphantom{space} \frac{2}{\log_2 3}|\)

Now we have to multiply both determinants to find the value of the expression: \(D_1 \times D_2 = (5 \log_2 3 - 1) \cdot \log_2 3 (\log_2 3 - 1)\) \((5 \log_2 3 - 1)(\log_2 3 - 1) = \log_2 3(5 \log_2 3 - 1) - 1(5 \log_2 3 - 1)\) \(5 \log_2^3 3 - \log_2^2 3 - 5 \log_2 3 + 1 = 10\) The value of the given expression is 10, which corresponds to the option (c).