Question

Value of the determinant \(\begin{vmatrix} \log_{3}512 & \log_{4}3 \\ \log_{3}8 & \log_{4}9 \end{vmatrix}\) is:

• A. \(15\)
• B. \( \frac{15}{2} \)
• C. \(21\)
• D. \( \frac{21}{2} \)

Hint:

Convert all numbers into prime powers to simplify logarithmic determinants quickly.

Answer 1

The Correct Option is B

Concept: Use logarithmic identities and properties of determinants: \[ \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc \]

Step 1: {Apply determinant formula
\[ D = (\log_3 512)(\log_4 9) - (\log_3 8)(\log_4 3) \]

Step 2: {Simplify logarithms
\[ 512 = 2^9,\quad 8 = 2^3,\quad 9 = 3^2 \] \[ \log_3 512 = 9\log_3 2,\quad \log_3 8 = 3\log_3 2 \] \[ \log_4 9 = 2\log_4 3 \]

Step 3: {Substitute values
\[ D = (9\log_3 2)(2\log_4 3) - (3\log_3 2)(\log_4 3) \] \[ D = 18\log_3 2\log_4 3 - 3\log_3 2\log_4 3 \] \[ D = 15\log_3 2\log_4 3 \] Using identity: \[ \log_3 2 \cdot \log_4 3 = \frac{1}{2} \] \[ D = \frac{15}{2} \] \fbox{\( \frac{15}{2} \)