Question

If a, b, c are non-zero and different from 1, then the value of $\left|\begin{array}{ccc}{\log }_{a}1& {\log }_{a}b& {\log }_{a}c\\ {\log }_a\left(\frac{1}{b}\right)& {\log }_{b}1& {\log }_a\left(\frac{1}{c}\right)\\ {\log }_a\left(\frac{1}{c}\right)& {\log }_{a}c& {\log }_{c}1\end{array}\right|$ 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)

$\left|\begin{array}{ccc}{\log }_{a}1& {\log }_{a}b& {\log }_{a}c\\ {\log }_a\left(\frac{1}{b}\right)& {\log }_{b}1& {\log }_a\left(\frac{1}{c}\right)\\ {\log }_a\left(\frac{1}{c}\right)& {\log }_{a}c& {\log }_{c}1\end{array}\right|$
= $\left|\begin{array}{ccc}\log 1& \log b& \log c\\ \log b& \log 1& 1\log c\\ 1logc& logc& 0\end{array}\right|$
$=\left|\begin{array}{ccc}0& \log b& \log c\\ \log b& 0& 1\log c\\ 1\log c& \log c& 0\end{array}\right|=0$