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