\({\log _{0,5}}\left( {{{\log }_2}\left( {2x-1} \right)} \right) \ge -1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\log _{0,5}}\left( {{{\log }_2}\left( {2x-1} \right)} \right) \ge {\log _{0,5}}2\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,0 < {\log _2}\left( {2x-1} \right) \le 2\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\log _2}1 < {\log _2}\left( {2x-1} \right) \le {\log _2}4\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,1 < 2x-1 \le 4\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,2 < 2x \le 5\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,1 < x \le 2,5\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x\, \in \,\,\left( {1;2,5} \right].\)
Ответ: \(\,\left( {1;2,5} \right].\)