Question

If $x$ $y$ $z$ are $\ne 0,\ne 1$ the value of $\left|\begin{array}{ccc}\log x& \log y& \log z\\ \log 2x& \log 2y& \log 2z\\ \log 3x& \log 3y& \log 3z\end{array}\right|$ is equal to

• A. $\log \left(xyz\right)$
• B. $\log \left(6xyz\right)$
• C. $0$
• D. $\log (x+y+z)$

Answer 1

The correct option is C $0$

Given $\left|\begin{array}{ccc}\log x& \log y& \log z\\ \log 2x& \log 2y& \log 2z\\ \log 3x& \log 3y& \log 3z\end{array}\right|$

$=\left|\begin{array}{ccc}\log x& \log y& \log z\\ \log 2+\log x& \log 2+\log y& \log 2+\log z\\ \log 3+\log x& \log 3+\log y& \log 3+\log z\end{array}\right|$, $[\because log(ab)=\log a+\log b]$

$=\left|\begin{array}{ccc}\log x& \log y& \log z\\ \log 2& \log 2& \log 2\\ \log 3& \log 3& \log 3\end{array}\right|$ , use $R_3\to R_3-R_2$ and $R_2\to R_2-R_1$

$=(\log 2\cdot \log 3)\left|\begin{array}{ccc}\log x& \log y& \log z\\ 1& 1& 1\\ 1& 1& 1\end{array}\right|$, take $\log 2$ and $\log 3$ common from second and third row respectively

$=(\log 2\cdot \log 3)\left(0\right)$, since two rows of the above determinant are same (2nd and 3rd)

$=0$