The value of left|begin{array}{ccc}1 & log _{ | 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}

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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 write the elements as:$\Delta = \left|\begin{array}{ccc}1 & \frac{\ln y}{\ln x} & \frac{\ln z}{\ln x} \ \frac{\ln x}{\ln y} & 1 & \frac{\ln z}{\ln y} \ \frac{\ln x}{\ln z} & \frac{\ln y}{\ln z} & 1\end{array}ight|$.Multiply $R_1$ by $\ln x$,$R_2$ by $\ln y$,and $R_3$ by $\ln z$:$\Delta = \frac{1}{\ln x \ln y \ln z} \left|\begin{array}{ccc}\ln x & \ln y & \ln z \ \ln x & \ln y & \ln z \ \ln x & \ln y & \ln z\end{array}ight|$.Since all three rows are identical,the value of the determinant is $0$.

The value of $\left|\begin{array}{ccc}1 & \log _{x} y & \log _{x} z \\ \log _{y} x & 1 & \log _{y} z \\ \log _{z} x & \log _{z} y & 1\end{array}\right|$ is equal to

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