Question

For positive numbers $x,y,z$ the numerical value of the determinant $\left|\begin{array}{ccc}1& {\log }_{x}y& {\log }_{x}z\\ {\log }_{y}x& 3& {\log }_{y}z\\ {\log }_{z}x& {\log }_{z}y& 5\end{array}\right|$ is

• A. $0$
• B. $\log x\log y\log z$
• C. $1$
• D. $8$

Answer 1

The correct option is D $8$

$\left|\begin{array}{ccc}1& lo{g}_{x}y& lo{g}_{x}z\\ lo{g}_{y}x& 3& lo{g}_{y}z\\ lo{g}_{z}x& lo{g}_{z}y& 5\end{array}\right|$
$=\left|\begin{array}{ccc}1& \frac{{\log }_{e}y}{{\log }_{e}x}& \frac{{\log }_{e}z}{{\log }_{e}x}\\ \frac{{\log }_{e}x}{{\log }_{e}y}& 3& \frac{{\log }_{e}z}{{\log }_{e}y}\\ \frac{{\log }_{e}x}{{\log }_{e}z}& \frac{{\log }_{e}y}{{\log }_{e}z}& 5\end{array}\right|$
$=\frac{1}{{\log }_{e}x{\log }_{e}y{\log }_{e}z}\left|\begin{array}{ccc}{\log }_{e}x& {\log }_{e}y& {\log }_{e}z\\ {\log }_{e}x& 3{\log }_{e}y& {\log }_{e}z\\ {\log }_{e}x& {\log }_{e}y& 5{\log }_{e}z\end{array}\right|$
$=\left|\begin{array}{ccc}1& 1& 1\\ 1& 3& 1\\ 1& 1& 5\end{array}\right|$
$=14-4-2=8$