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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
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\( 00 \)
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 $R_{3} \rightarrow R_{3}-R_{1}$
$\mid \begin{array}{ll}\log x & \log y & \log z \\ \log 2 & \log 2 & \log 2 \\ \log 3 & \log 3 & \log 3 & \\ = & \log 2 \log 3 \\ \text { When the two rows of determinants are identical the value of determinant is equal to } 0 & \begin{array}{ccc}\log x & \log y & \log z \\ 1 & 1 & 1 \\ 1 & 1 & 1\end{array} \mid\end{array}$
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 $R_{3} \rightarrow R_{3}-R_{1}$
$\mid \begin{array}{ll}\log x & \log y & \log z \\ \log 2 & \log 2 & \log 2 \\ \log 3 & \log 3 & \log 3 & \\ = & \log 2 \log 3 \\ \text { When the two rows of determinants are identical the value of determinant is equal to } 0 & \begin{array}{ccc}\log x & \log y & \log z \\ 1 & 1 & 1 \\ 1 & 1 & 1\end{array} \mid\end{array}$
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