Задача 24. Решите уравнение \(\dfrac{4}{3}\log _3^2{\left( {5x-6} \right)^3}-{\log _3}{\left( {5x-6} \right)^3} \cdot {\log _3}{x^6} = -6\,\,\log _3^2\dfrac{1}{x}\)
ОТВЕТ: \(\dfrac{{36}}{{25}};\;\;\dfrac{3}{2}.\)
\(\dfrac{4}{3}\log _3^2{\left( {5x-6} \right)^3}-{\log _3}{\left( {5x-6} \right)^3} \cdot {\log _3}{x^6} = -6\log _3^2\dfrac{1}{x}.\) Запишем ОДЗ: \(\left\{ {\begin{array}{*{20}{c}}{{{\left( {5x-6} \right)}^3} > 0,}\\{{x^6} > 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\dfrac{1}{x} > 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x > \dfrac{6}{5},}\\{x \ne 0,}\\{x > 0\,\,\,}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x \in \left( {\dfrac{6}{5};\infty } \right).\) \(\dfrac{4}{3} \cdot {\left( {3{{\log }_3}\left( {5x-6} \right)} \right)^2}-18{\log _3}\left( {5x-6} \right) \cdot {\log _3}x + 6 \cdot {\left( {-{{\log }_3}x} \right)^2} = 0\,\,\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\,12\log _3^2\left( {5x-6} \right)-18{\log _3}\left( {5x-6} \right) \cdot {\log _3}x + 6\log _3^2x = 0.\) Так как \(x = 1\) не является корнем уравнения, то разделим обе части последнего уравнения на: \(6\log _3^2x\). \(2 \cdot {\left( {\dfrac{{{{\log }_3}\left( {5x-6} \right)}}{{{{\log }_3}x}}} \right)^2}-3 \cdot \dfrac{{{{\log }_3}\left( {5x-6} \right)}}{{{{\log }_3}x}} + 1 = 0.\) Пусть \(\dfrac{{{{\log }_3}\left( {5x-6} \right)}}{{{{\log }_3}x}} = t\). Тогда: \(2{t^2}-3t + 1 = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{t = 1,}\\{t = \dfrac{1}{2}.}\end{array}} \right.\) Возвращаясь к прежней переменной, получим: \(\left[ {\begin{array}{*{20}{c}}{\dfrac{{{{\log }_3}\left( {5x-6} \right)}}{{{{\log }_3}x}} = 1,}\\{\dfrac{{{{\log }_3}\left( {5x-6} \right)}}{{{{\log }_3}x}} = \dfrac{1}{2}}\end{array}\,\,\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{{{\log }_3}\left( {5x-6} \right) = {{\log }_3}x,}\\{2{{\log }_3}\left( {5x-6} \right) = {{\log }_3}x}\end{array}\,\,\,\,\,\,\,\,\, \Leftrightarrow } \right.} \right.\) \( \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{5x-6 = x,\,\,\,}\\{{{\left( {5x-6} \right)}^2} = x}\end{array}\,\,\,\,\,\,} \right. \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = \dfrac{3}{2},\,\,}\\{x = 1,\,\,\,\,\,}\\{x = \dfrac{{36}}{{25}}.}\end{array}} \right.\,\,\) Корень \(x = 1\) не удовлетворяет ОДЗ. Ответ: \(\dfrac{{36}}{{25}};\,\,\,\dfrac{3}{2}\).