\(\sqrt {{{\log }_3}{x^9}} -4{\log _9}\sqrt {3x} = 1\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\sqrt {9{{\log }_3}x} -{\log _3}\left( {3x} \right) = 1\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,3\sqrt {{{\log }_3}x} -1-{\log _3}x = 1.\)
Пусть \(\sqrt {{{\log }_3}x} = t\), где \(t \ge 0\). Тогда уравнение примет вид:
\(3t-1-{t^2} = 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{t^2}-3t + 2 = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{t = 1,}\\{t = 2.}\end{array}} \right.\)
Возвращаясь к прежней переменной, получим:
\(\left[ {\begin{array}{*{20}{c}}{\sqrt {{{\log }_3}x} = 1,}\\{\sqrt {{{\log }_3}x} = 2}\end{array}\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\,} \right.\left[ {\begin{array}{*{20}{c}}{{{\log }_3}x = 1,}\\{{{\log }_3}x = 4}\end{array}\,\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = 3,\,\,\,}\\{x = 81.}\end{array}} \right.} \right.\)
Ответ: 3; 81.