\({\log _4}\left( {6 + 5x} \right) = {\log _4}\left( {3 + x} \right) + 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\log _4}\left( {6 + 5x} \right) = {\log _4}\left( {3 + x} \right) + {\log _4}4\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,{\log _4}\left( {6 + 5x} \right) = {\log _4}\left( {12 + 4x} \right)\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{6 + 5x > 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{6 + 5x = 12 + 4x}\end{array}\,\,\,\,\,\,\, \Leftrightarrow } \right.\)
\( \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x > -\dfrac{6}{5},}\\{x = 6\,\,\,\,\,\,\,}\end{array}\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,x = 6.} \right.\)
Ответ: 6.