b
(b) \(\log x:\log y:\log z = y - z:z - x:x - y\)

\( \Rightarrow \) \({{\log x} \over {y - z}} = {{\log y} \over {z - x}} = {{\log z} \over {x - y}} = k\,{\rm{(say)}}\)

\( \Rightarrow \) \(\log x = k(y - z),\,\log y = k(z - x),\,\log z = k(x - y)\)

\(\therefore \log x + \log y + \log z = 0\)\( \Rightarrow \)\(x + y = 1\) \( \Rightarrow \)\(xyz = 1\).

\(x\log x + y\log y + z\log z\)

=\(x.k.(y - z) + y.k.(z - x) + z.k(x - y) = 0\)

\( \Rightarrow \)\(\log ({x^x}.{y^y}.{z^z}) = \log 1\)

\(\therefore x^xy^yz^z=1 \).