\({\log _{\sqrt 3 }}x + {\log _{\sqrt[3]{3}}}x-{\log _{\sqrt[6]{3}}}x \ge -2\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,2{\log _3}x + 3{\log _3}x-6{\log _3}x \ge -2\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,-{\log _3}x \ge -2\left| { \cdot \left( {-1} \right)\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{{\log }_3}x \le 2\,\,\,\,\,\,\, \Leftrightarrow } \right.\)
\( \Leftrightarrow \,\,\,\,\,\,\,{\log _3}x \le {\log _3}9\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \le 9,}\\{x > 0\,}\end{array}\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x\, \in \,\left( {0;9} \right].} \right.\)
Ответ: \(\left( {0;9} \right].\)