12В. а) Решите уравнение \({\lg ^2}\left( {8x-9} \right) = {\lg ^2}\left( {6x-4} \right)\);
б) Найдите все корни принадлежащие промежутку \(\left[ {1;\;2} \right]\).
ОТВЕТ: а) \(\frac{5}{2};\;\;\;\frac{7}{6};\) б) \(\frac{7}{6}.\)
а) \({\lg ^2}\left( {8x-9} \right) = {\lg ^2}\left( {6x-4} \right).\) Запишем ОДЗ: \(\left\{ {\begin{array}{*{20}{c}}{8x-9 > 0,}\\{6x-4 > 0\,}\end{array}} \right.\;\;\;\; \Leftrightarrow \;\;\;\;\left\{ {\begin{array}{*{20}{c}}{x > \frac{9}{8},}\\{x > \frac{2}{3}\,\,}\end{array}} \right.\;\;\;\; \Leftrightarrow \;\;\;\;x \in \left( {\frac{9}{8};\infty } \right).\) \({\lg ^2}\left( {8x-9} \right) = {\lg ^2}\left( {6x-4} \right)\;\;\;\; \Leftrightarrow \;\;\;\;{\lg ^2}\left( {8x-9} \right)-{\lg ^2}\left( {6x-4} \right) = 0\;\;\;\; \Leftrightarrow \) \( \Leftrightarrow \;\;\;\;\left( {\lg \left( {8x-9} \right)-\lg \left( {6x-4} \right)} \right)\left( {\lg \left( {8x-9} \right) + \lg \left( {6x-4} \right)} \right) = 0\;\;\;\; \Leftrightarrow \) \( \Leftrightarrow \;\;\;\;\left[ {\begin{array}{*{20}{c}}{\lg \left( {8x-9} \right)-\lg \left( {6x-4} \right) = 0,\,\,}\\{\lg \left( {8x-9} \right) + \lg \left( {6x-4} \right) = \lg 1\,}\end{array}} \right.\;\;\;\; \Leftrightarrow \;\;\;\;\left[ {\begin{array}{*{20}{c}}{\lg \left( {8x-9} \right) = \lg \left( {6x-4} \right),}\\{\left( {8x-9} \right)\left( {6x-4} \right) = 1\,\,\,\,\,\,\,\,}\end{array}} \right.\;\;\;\; \Leftrightarrow \) \( \Leftrightarrow \;\;\;\;\left[ {\begin{array}{*{20}{c}}{8x-9 = 6x-4,\;\,\,\,\,\,\,\,\;\;\;\;\;\;\;\;\;\;\;\;\,\;}\\{48{x^2}-32x-54x + 36-1 = 0}\end{array}} \right.\;\;\;\; \Leftrightarrow \;\;\;\;\left[ {\begin{array}{*{20}{c}}{2x-5 = 0,\;\;\;\;\;\;\;\;\;\;\;\;\,\;}\\{48{x^2}-86x + 35 = 0}\end{array}} \right.\;\;\;\; \Leftrightarrow \) \( \Leftrightarrow \;\;\;\;\left[ {\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{x = \frac{5}{2},\;\;\;\;\;\;\;\;\;\;\;\,\,}\\{x = \frac{7}{6},\;\;\;\;\;\,\,\;\;\;\;\;\;}\end{array}}\\{x = \frac{5}{8} \notin \left( {\frac{9}{8};\infty } \right)}\end{array}} \right.\;\;\;\; \Leftrightarrow \;\;\;\;\left[ {\begin{array}{*{20}{c}}{x = \frac{5}{2},}\\{x = \frac{7}{6}.\,}\end{array}} \right.\) б) Отберём корни, принадлежащие отрезку \(\left[ {1;\;2} \right].\) \(x = \frac{5}{2} \notin \left[ {1;\;2} \right];\) \(x = \frac{7}{6}\,\, \in \,\left[ {1;\;2} \right].\) Ответ: а) \(\frac{5}{2};\;\;\;\frac{7}{6};\) б) \(\frac{7}{6}.\)