Простейшие логарифмические уравнения. Задача 20

Задача 20. Решите уравнение \({\log _{x + 3}}81 = 4\)

Ответ

ОТВЕТ: 0.

Решение

\({\log _{x + 3}}81 = 4.\)

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

\({\log _{x + 3}}81 = 4\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\log _{x + 3}}{3^4} = 4\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,4{\log _{x + 3}}3 = 4\,\,\,\,\,\,\, \Leftrightarrow \)

\( \Leftrightarrow \,\,\,\,\,\,\,{\log _{x + 3}}3 = 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x + 3 = 3\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = 0.\)

Ответ: 0.