Question

The value of \[ \left| \begin{array}{ccc} \log x & \log y & \log z \log ax & \log ay & \log az \log bx & \log by & \log bz \end{array} \right| \] is

• A. \(\log ab\)
• B. \(\log(abxyz)\)
• C. \(1\)
• D. \(0\)

Hint:

If any two rows (or columns) of a determinant are proportional or identical, then \[ \boxed{\text{Determinant}=0.} \]

Answer 1

The Correct Option is D

Step 1: Use the logarithmic identity. Using \[ \log(ax)=\log a+\log x, \] the determinant becomes \[ \left| \begin{array}{ccc} \log x & \log y & \log z \log a+\log x & \log a+\log y & \log a+\log z \log b+\log x & \log b+\log y & \log b+\log z \end{array} \right|. \]

Step 2: Apply row operations. Perform \[ R_2\rightarrow R_2-R_1,\qquad R_3\rightarrow R_3-R_1. \] Then, \[ \left| \begin{array}{ccc} \log x & \log y & \log z \log a & \log a & \log a \log b & \log b & \log b \end{array} \right|. \] Since the second and third rows each have identical entries, they are proportional. Therefore, the determinant is \[ \boxed{0.} \] Hence, \[ \boxed{(D)} \] is the correct answer.