Задача 14. Решите уравнение \(\dfrac{2}{{\sqrt 3 {{\log }_2}\sqrt {{x^2}} }}-\dfrac{1}{{\sqrt {{{\log }_2}\left( {-x} \right)} }} = 0\)
ОТВЕТ: \(-\sqrt[3]{{16}}.\)
\(\dfrac{2}{{\sqrt 3 {{\log }_2}\sqrt {{x^2}} }}-\dfrac{1}{{\sqrt {{{\log }_2}\left( {-x} \right)} }} = 0.\) Запишем ОДЗ: \(\left\{ {\begin{array}{*{20}{c}}{{{\log }_2}\sqrt {{x^2}} \ne 0,\,\,\,\,\,\,}\\{{x^2} > 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{-x > 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\sqrt {{{\log }_2}\left( {-x} \right)} \ne 0\,\,\,\,\,}\end{array}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \ne \pm 1,}\\{x \ne 0,\,\,}\\{x < 0,\,\,}\\{x \ne -1\,\,}\end{array}\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,x\, \in \,\left( {-\infty ;-1} \right) \cup \left( {-1;0} \right).} \right.} \right.\) \(\dfrac{2}{{\sqrt 3 {{\log }_2}\sqrt {{x^2}} }}-\dfrac{1}{{\sqrt {{{\log }_2}\left( {-x} \right)} }} = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\dfrac{2}{{\sqrt 3 {{\log }_2}\left| x \right|}}-\dfrac{1}{{\sqrt {{{\log }_2}\left( {-x} \right)} }} = 0.\) Так как \(x\, \in \,\left( {-\infty ;-1} \right) \cup \left( {-1;0} \right)\), то \(\left| x \right| = -x\). Тогда уравнение примет вид: \(\frac{2}{{\sqrt 3 {{\log }_2}\left( {-x} \right)}}-\dfrac{1}{{\sqrt {{{\log }_2}\left( {-x} \right)} }} = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\sqrt 3 {\log _2}\left( {-x} \right) = 2\sqrt {{{\log }_2}\left( {-x} \right)} \,\,\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\,\sqrt 3 {\left( {\sqrt {{{\log }_2}\left( {-x} \right)} } \right)^2}-2\sqrt {{{\log }_2}\left( {-x} \right)} = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\sqrt {{{\log }_2}\left( {-x} \right)} \cdot \left( {\sqrt 3 \cdot \sqrt {{{\log }_2}\left( {-x} \right)} -2} \right) = 0\,\,\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\sqrt {{{\log }_2}\left( {-x} \right)} = 0,}\\{\sqrt {3{{\log }_2}\left( {-x} \right)} = 2}\end{array}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{{{\log }_2}\left( {-x} \right) = 0,}\\{{{\log }_2}\left( {-x} \right) = \dfrac{4}{3}}\end{array}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = -1,\,\,\,\,\,\,\,\,}\\{x = -2\sqrt[3]{2}.}\end{array}} \right.} \right.} \right.\) Корень \(x = -1\) не удовлетворяет ОДЗ. Ответ: \(-2\sqrt[3]{2}\).