For positive numbers x, y and z,the numerical | vedclass

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(A) Let the determinant be $\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{arra

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(A) Let the determinant be $\Delta = \left| \begin{array}{ccc} 1 & \log_x y & \log_x z \ \log_y x & 1 & \log_y z \ \log_z x & \log_z y & 1 \end{array} ight|$.Using the property $\log_a b = \frac{\ln b}{\ln a}$,we can rewrite the elements as:$\Delta = \left| \begin{array}{ccc} 1 & \frac{\ln y}{\ln x} & \frac{\ln z}{\ln x} \ \frac{\ln x}{\ln y} & 1 & \frac{\ln z}{\ln y} \ \frac{\ln x}{\ln z} & \frac{\ln y}{\ln z} & 1 \end{array} ight|$.Taking $\frac{1}{\ln x}$ common from $R_1$,$\frac{1}{\ln y}$ from $R_2$,and $\frac{1}{\ln z}$ from $R_3$ is not directly helpful,so let's multiply $R_1$ by $\ln x$,$R_2$ by $\ln y$,and $R_3$ by $\ln z$ and divide the determinant by $(\ln x \ln y \ln z)$:$\Delta = \frac{1}{\ln x \ln y \ln z} \left| \begin{array}{ccc} \ln x & \ln y & \ln z \ \ln x & \ln y & \ln z \ \ln x & \ln y & \ln z \end{array} ight|$.Since all three rows are identical,the value of the determinant is $0$.

For positive numbers $x, y$ 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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