Задача 17. Найдите корень уравнения \({\log _7}\left( {8-2x} \right)-{\log _7}8 = {\log _7}\dfrac{1}{{40}}\).
Решение
\({\log _7}\left( {8-2x} \right)-{\log _7}8 = {\log _7}\dfrac{1}{{40}}\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\,{\log _7}\dfrac{{8-2x}}{8} = {\log _7}\dfrac{1}{{40}}\,\,\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,\dfrac{{8-2x}}{8} = \dfrac{1}{{40}}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,8-2x = \dfrac{1}{5}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,-2x = -7,8\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,x = 3,9.\)
Ответ: 3,9.