If \( a, b, c \) are positive real numbers each distinct from unity, then the value of the determinant
\[
\begin{vmatrix}
1 & \log_a b & \log_a c \\
\log_b a & 1 & \log_b c \\
\log_c a & \log_c b & 1
\end{vmatrix}
\]
is:
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For log determinants:
Convert logs using \( \log_a b = \frac{\ln b}{\ln a} \).
Try row scaling to expose symmetry.
Identical or proportional rows \( \Rightarrow \) determinant = 0.