Value of the determinant \( \begin{vmatrix}\log_3 512 & \log_4 3\\\log_3 8 & \log_4 9\end{vmatrix} \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 15/2
Since \( 512=2^9 \) and \( 8=2^3 \), \( \log_3 512=9\log_3 2 \) and \( \log_3 8=3\log_3 2 \). Also \( \log_4 3=\frac{1}{2}\log_2 3 \) and \( \log_4 9=\log_2 3 \). So
\( \displaystyle \Delta=9\log_3 2\cdot\log_2 3-\frac{1}{2}\log_2 3\cdot 3\log_3 2 \)
Since \( \log_3 2\cdot\log_2 3=1 \):
\( \displaystyle \Delta=9-\frac{3}{2} \)
\( \displaystyle =\frac{15}{2} \)