Простейшие логарифмические уравнения. Задача 19

Задача 19. Решите уравнение \({\log _{x + 6}}32 = 5\)

Ответ

ОТВЕТ: \(-4\).

Решение

\({\log _{x + 6}}32 = 5.\)

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

\({\log _{x + 6}}32 = 5\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\left( {x + 6} \right)^5} = 32\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\left( {x + 6} \right)^5} = {2^5}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x + 6 = 2\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = -4.\)

Ответ: \(-4\).