Q.Find the value of the product: $\begin{vmatrix} \log_3 64 & \log_4 3 \\ \log_3 8 & \log_4 9 \end{vmatrix} \times \begin{vmatrix} \log_2 3 & \log_8 3 \\ \log_3 4 & \log_3 4 \end{vmatrix}$.

Write every logarithm in natural-log form $\log_b x = \dfrac{\ln x}{\ln b}$ and simplify.

Step 1 — first determinant $D_1 = \begin{vmatrix}\log_3 64 & \log_4 3\\ \log_3 8 & \log_4 9\end{vmatrix}$.

Convert: $\log_3 64 = \dfrac{6\ln2}{\ln3}$, $\log_4 9 = \dfrac{2\ln3}{2\ln2}=\dfrac{\ln3}{\ln2}$, $\log_4 3 = \dfrac{\ln3}{2\ln2}$, $\log_3 8 = \dfrac{3\ln2}{\ln3}$.

Step 2 — second determinant $D_2 = \begin{vmatrix}\log_2 3 & \log_8 3\\ \log_3 4 & \log_3 4\end{vmatrix}$.

$D_2 = \log_3 4\,(\log_2 3 - \log_8 3).$ …

Change-of-base to simplify log entries, evaluate each 2×2 d …

• Change-of-base slips, e.g. writing $\log_4 9 = \log_2 3$ without checking $\dfrac{2\ln3}{2\ln2}$. …