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(A) Let the determinant be $D = \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}

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$0$

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(A) Let the determinant be $D = \left| \begin{array}{ccc} 1 & \log_x y & \log_x z \ \log_y x & 1 & \log_y z \ \log_z x & \log_z y & 1 \end{array} ight|$.Using the change of base formula $\log_a b = \frac{\log b}{\log a}$,we can rewrite the determinant as:$D = \left| \begin{array}{ccc} 1 & \frac{\log y}{\log x} & \frac{\log z}{\log x} \ \frac{\log x}{\log y} & 1 & \frac{\log z}{\log y} \ \frac{\log x}{\log z} & \frac{\log y}{\log z} & 1 \end{array} ight|$.Now,multiply $R_1$ by $\log x$,$R_2$ by $\log y$,and $R_3$ by $\log z$:$D = \frac{1}{(\log x)(\log y)(\log z)} \left| \begin{array}{ccc} \log x & \log y & \log z \ \log x & \log y & \log z \ \log x & \log y & \log z \end{array} ight|$.Since all three rows are identical,the value of the determinant is $0$.

For positive numbers $x, y, \text{ and } z$,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

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