Question

If a, b, c are non-zero and different from 1, then the value of $\begin{vmatrix} \log_a1&\log_ab &\log_ac \\[0.3em] \log_a\left(\frac{1}{b}\right) & \log_b1 & \log_a\left(\frac{1}{c}\right) \\[0.3em] \log_a\left(\frac{1}{c}\right) & \log_ac & \log_c1 \end{vmatrix}$ is

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

Answer 1

Answer: (A)

$\begin{vmatrix} \log_a1&\log_ab &\log_ac \\[0.3em] \log_a\left(\frac{1}{b}\right) & \log_b1 & \log_a\left(\frac{1}{c}\right) \\[0.3em] \log_a\left(\frac{1}{c}\right) & \log_ac & \log_c1 \end{vmatrix}$
= $\begin{vmatrix} \log\,1&\log\,b &\log\,c \\[0.3em] \log\, b& \log\, 1 &1\, \log\, c\\[0.3em] 1\, log\, c & log \, c & 0 \end{vmatrix}$
$ = \begin{vmatrix}0&\log\,b &\log\,c \\[0.3em] \log\, b& 0 &1\, \log\, c\\[0.3em] 1\,\log\, c & \log \, c & 0 \end{vmatrix} = 0 $