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(A) Let the determinant be $\Delta = \left|\begin{array}{ccc}1 & \log _a b & \log _a c \ \log _b a & 1 & \log _b c \ \log _c a & \log _c b & 1\end{a

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(A) Let the determinant be $\Delta = \left|\begin{array}{ccc}1 & \log _a b & \log _a c \ \log _b a & 1 & \log _b c \ \log _c a & \log _c b & 1\end{array}ight|$.Using the property $\log _x y = \frac{\ln y}{\ln x}$, we can rewrite the elements as:$\log _a b = \frac{\ln b}{\ln a}$, $\log _a c = \frac{\ln c}{\ln a}$, $\log _b a = \frac{\ln a}{\ln b}$, $\log _b c = \frac{\ln c}{\ln b}$, $\log _c a = \frac{\ln a}{\ln c}$, $\log _c b = \frac{\ln b}{\ln c}$.Substituting these into the determinant:$\Delta = \left|\begin{array}{ccc}1 & \frac{\ln b}{\ln a} & \frac{\ln c}{\ln a} \ \frac{\ln a}{\ln b} & 1 & \frac{\ln c}{\ln b} \ \frac{\ln a}{\ln c} & \frac{\ln b}{\ln c} & 1\end{array}ight|$.Factor out $\frac{1}{\ln a}$ from $R_1$, $\frac{1}{\ln b}$ from $R_2$, and $\frac{1}{\ln c}$ from $R_3$:$\Delta = \frac{1}{\ln a \ln b \ln c} \left|\begin{array}{ccc}\ln a & \ln b & \ln c \ \ln a & \ln b & \ln c \ \ln a & \ln b & \ln c\end{array}ight|$.Since all three rows are identical, the value of the determinant is $0$.

If $a, b, c$ are positive real numbers each distinct from unity, then the value of the determinant $\left|\begin{array}{ccc}1 & \log _a b & \log _a c \\ \log _b a & 1 & \log _b c \\ \log _c a & \log _c b & 1\end{array}\right|$ is

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