c
(c) \({\log _{0.04}}(x - 1) \ge {\log _{0.2}}(x - 1)\) …..\((i)\)

For log to be defined \(x - 1 > 0 \Rightarrow x > 1\)

From \((i),\) \({\log _{{{(0.2)}^2}}}(x - 1) \ge {\log _{0.2}}(x - 1)\)

\( \Rightarrow \)\({1 \over 2}{\log _{0.2}}(x - 1) \ge {\log _{0.1}}(x - 1)\)

\( \Rightarrow \)\(\sqrt {x - 1} \le (x - 1)\)

\( \Rightarrow \)\(\sqrt {x - 1} (1 - \sqrt {x - 1} ) \le 0\) \( \Rightarrow \)\(1 - \sqrt {x - 1} \le 0\)

\( \Rightarrow \)\(\sqrt {x - 1} \ge 1\) \( \Rightarrow \)\(x \ge 2\),

\(\therefore \,\,x \in [2,\,\infty )\) .