\({\log _2}\left( {x + 3} \right) = {\log _2}\left( {4 + x} \right)-1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\log _2}\left( {x + 3} \right) + {\log _2}2 = {\log _2}\left( {4 + x} \right)\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,{\log _2}\left( {2x + 6} \right) = {\log _2}\left( {4 + x} \right)\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{2x + 6 > 0,\,\,\,\,\,\,\,}\\{2x + 6 = 4 + x}\end{array}\,\,\,\,\,\,\,\,\, \Leftrightarrow } \right.\,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x > -3,}\\{x = -2}\end{array}\,\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,x = -2.} \right.\)
Ответ: \(-2\).