График показательной функции \(f\left( x \right) = {a^x} + b\) проходит через точки \(\left( {0; — 1} \right)\) и \(\left( {2;1} \right)\). Следовательно:
\(\left\{ {\begin{array}{*{20}{c}}{ — 1 = {a^0} + b}\\{1 = {a^2} + b}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{b = — 2\,\,\,\,\,\,}\\{1 = {a^2} + b}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\,{a^2} = 3\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,a = \sqrt 3 .\)
Таким образом: \(f\left( x \right) = {\sqrt 3 ^x} — 2\) и \({\sqrt 3 ^x} — 2 = 25\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,{3^{\frac{x}{2}}} = 27\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\,x = 6.\)
Ответ: 6.