The value of \( \begin{vmatrix}log_3 512 & log_4 3\\log_3 8 & log_4 9\end{vmatrix} \) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \displaystyle\frac{15}{2} \)
Since \( 512=2^9 \), \( 8=2^3 \), \( \log_4 3=\frac{1}{2}\log_2 3 \) and \( \log_4 9=\log_2 3 \), every entry is a multiple of \( \log_3 2 \) or \( \log_2 3 \):
\( \displaystyle \log_3 512=9\log_3 2,\qquad \log_3 8=3\log_3 2 \)
\( \displaystyle \Delta=9\log_3 2\cdot\log_2 3-\frac{1}{2}\log_2 3\cdot 3\log_3 2 \)
Since \( \log_3 2\cdot\log_2 3=1 \) (change of base),
\( \displaystyle \Delta=9-\frac{3}{2} \)
\( \displaystyle =\frac{15}{2} \)