\({\log _{x + 6}}32 = 5\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{{{\left( {x + 6} \right)}^5} = 32}\\{x + 6 > 0\,\,\,\,\,\,\,\,\,\,}\\{x + 6 \ne 1\,\,\,\,\,\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{{{\left( {x + 6} \right)}^5} = {2^5}}\\{x + 6 > 0\,\,\,\,\,\,\,\,}\\{x + 6 \ne 1\,\,\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x + 6 = 2}\\{x + 6 > 0}\\{x + 6 \ne 1}\end{array}\,\,\,\,\,\, \Leftrightarrow } \right.\,\,\,\,\,\,\,\,x + 6 = 2\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = — 4.\)
Ответ: – 4.