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

We have

$\begin{array}{l}\left|\begin{array}{lll}1& \frac{\log y}{\log x}& \frac{\log z}{\log x}\\ \frac{\log x}{\log y}& 1& \frac{1ogz}{\log y}\\ \frac{\log x}{\log z}& \frac{\log y}{\log z} & 1\end{array}\right|\\ \end{array}=\frac{1}{\log x\log y\log z}\left|\begin{array}{lll}\log x& \log y& \log z\\ \log x& \log y& \log z\\ \log x& \log y& \log z\end{array}\right|=0$