График логарифмической функции \(f\left( x \right) = {\log _a}\left( {x + b} \right)\) проходит через точки \(\left( { — 1; — 1} \right)\) и \(\left( { — 3;0} \right)\). Следовательно:
\(\left\{ {\begin{array}{*{20}{c}}{ — 1 = {{\log }_a}\left( { — 1 + b} \right)}\\{0 = {{\log }_a}\left( { — 3 + b} \right)}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{{a^{ — 1}} = b — 1}\\{ — 3 + b = 1}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{a = \frac{1}{{b — 1}}}\\{b = 4\,\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,a = \frac{1}{3}.\)
Таким образом: \(f\left( x \right) = {\log _{\frac{1}{3}}}\left( {x + 4} \right)\) и \({\log _{\frac{1}{3}}}\left( {x + 4} \right) = — 5\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x + 4 = 243\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,x = 239.\)
Ответ: 239.
Задача 16. На рисунке изображён график функции \(f\left( x \right) = {\log _a}\left( {x + b} \right).\) Найдите значение x, при котором \(f\left( x \right) = — 5.\)