Простейшие логарифмические уравнения. Задача 21

Задача 21. Решите уравнение \({\log _{2x + 6}}8 = 3\)

Ответ

ОТВЕТ: \(-2\).

Решение

\({\log _{2x + 6}}8 = 3.\)

ОДЗ: \(\left\{ {\begin{array}{*{20}{c}}{2x + 6 > 0,}\\{2x + 6 \ne 1\,\,}\end{array}\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x > -3,\,\,\,\,}\\{x \ne -2,5}\end{array}\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x\, \in \,\left( {-3;-2,5} \right) \cup \left( {-2,5;\infty } \right).} \right.} \right.\)

\({\log _{2x + 6}}8 = 3\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\log _{2x + 6}}{2^3} = 3\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,3{\log _{2x + 6}}2 = 3\,\,\,\,\,\,\, \Leftrightarrow \)

\( \Leftrightarrow \,\,\,\,\,\,\,{\log _{2x + 6}}2 = 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,2x + 6 = 2\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = -2.\)

Ответ: \(-2\).