If \( x y z \) are not equal and \( \neq 0, \neq 1 \) the value of \( \left|\begin{array}{ccc}\log x & \log y & \log z \\ \log 2 x & \log 2 y & \log 2 z \\ \log 3 x & \log 3 y & \log 3 z\end{array}\right| \) is equal to

Given that, \(x, y, z \neq 0 \neq 1\)
As we know, \(\log 2 x-\log x=\log 2\)
and \(\log 3 x-\log x=\log 3\)
Given that \(\left|\begin{array}{ccc}\log x & \log y & \log z \\ \log 2 x & \log 2 y & \log 2 z \\ \log 3 x & \log 3 y & \log 3 z\end{array}\right|\)
\(R_{2} \rightarrow R_{2}-R_{1}\) and \(R_{3} \rightarrow R_{3}-R_{1}\)
\(\mid \begin{array}{ll}\log x & \log y & \log z \\ \log 2 & \log 2 & \log 2 \\ \log 3 & \log 3 & \log 3 & \\ = & \log 2 \log 3 \\ \text { When the two rows of determinants are identical the value of determinant is equal to } 0 & \begin{array}{ccc}\log x & \log y & \log z \\ 1 & 1 & 1 \\ 1 & 1 & 1\end{array} \mid\end{array}\)