a
(a) \([{\log _b}a.{\log _c}a - {\log _a}a] + [{\log _a}b.{\log _c}b - {\log _b}b]\)
\( + [{\log _a}c{\log _b}c - {\log _c}c] = 0\)

==> \({1 \over {\ln a.\ln b.\ln c}}[{(\ln a)^3} + {(\ln b)^3} + {(\ln c)^3} - 3\ln a.\ln b.\ln c] = 0\)

==> \({(\ln a)^3} + {(\ln b)^3} + {(\ln c)^3} - 3\ln a.\ln b.\ln c = 0\)

==> \(\ln a + \ln b + \ln c = 0\)

==> \(\ln (abc) = ln 1\), \([{a^3} + {b^3} + {c^3} - 3abc = 0\)

==> \(a + b + c = 0]\),

\(\therefore abc = 1\).