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