Question

If \(a, b, c\) are non-zero and different from \(1\), then the value of \[ \begin{vmatrix} \log_a 1 & \log_a b & \log_a c \\ \log_b \left(\frac{1}{b}\right) & \log_b 1 & \log_b \left(\frac{1}{c}\right) \\ \log_c \left(\frac{1}{c}\right) & \log_c c & \log_c 1 \end{vmatrix} \] is:

• A. \( 0 \)
• B. \( 1 + \log_a (a + b + c) \)
• C. \( \log_a (ab + bc + ca) \)
• D. \( 1 \)
• E. \( \log_a (a + b + c) \)

Hint:

The change of base rule $\log_a b \cdot \log_b c = \log_a c$ is essentially a "chain rule" for logarithms. It simplifies multi-base expressions into a single base instantly.

Answer 1

The Correct Option is A