График логарифмической функции \(f\left( x \right) = {\log _a}\left( {x + b} \right)\) проходит через точки \(\left( {1;1} \right)\) и \(\left( {0;0} \right)\). Следовательно:
\(\left\{ {\begin{array}{*{20}{c}}{1 = {{\log }_a}\left( {1 + b} \right)}\\{0 = {{\log }_a}b\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{1 + b = a}\\{b = 1\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,a = 2.\)
Таким образом: \(f\left( x \right) = {\log _2}\left( {x + 1} \right)\) и \({\log _2}\left( {x + 1} \right) = 5\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x + 1 = 32\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,x = 31.\)
Ответ: 31.
Задача 14. На рисунке изображён график функции \(f\left( x \right) = {\log _a}\left( {x + b} \right).\) Найдите значение x, при котором \(f\left( x \right) = 5.\)