Question

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

Answer 1

Let $\Delta =\left|\begin{array}{ccc}1& {\log }_{x}y& {\log }_{x}z\\ {\log }_{y}x& 1& {\log }_{y}z\\ {\log }_{z}x& {\log }_{z}y& 1\end{array}\right|=\left|\begin{array}{ccc}1& \frac{\log y}{\log x}& \frac{\log z}{\log x}\\ \frac{\log x}{\log y}& 1& \frac{\log z}{\log y}\\ \frac{\log x}{\log z}& \frac{\log y}{\log z}& 1\end{array}\right|$
On dividing and multiplying $R_1,R_2,R_3$ by $\log x,\log y,\log z$ respectively
$\Delta =\frac{1}{\log x\log y\log z}\left|\begin{array}{ccc}\log x& \log y& \log z\\ \log x& \log y& \log z\\ \log x& \log y& \log z\end{array}\right|=0$