График логарифмической функции \(f\left( x \right) = {\log _a}\left( {x + b} \right)\) проходит через точки \(\left( {1; — 2} \right)\) и \(\left( { — 1;0} \right)\). Следовательно:
\(\left\{ {\begin{array}{*{20}{c}}{ — 2 = {{\log }_a}\left( {1 + b} \right)}\\{0 = {{\log }_a}\left( { — 1 + b} \right)}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{{a^{ — 2}} = 1 + b}\\{ — 1 + b = 1}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{\frac{1}{{{a^2}}} = 1 + b}\\{b = 2\,\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,a = \frac{1}{{\sqrt 3 }}.\)
Таким образом: \(f\left( x \right) = {\log _{\frac{1}{{\sqrt 3 }}}}\left( {x + 2} \right)\) и \({\log _{\frac{1}{{\sqrt 3 }}}}\left( {x + 2} \right) = — 8\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x + 2 = 81\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,x = 79.\)
Ответ: 79.
Задача 15. На рисунке изображён график функции \(f\left( x \right) = {\log _a}\left( {x + b} \right).\) Найдите значение x, при котором \(f\left( x \right) = — 8.\)