\(81 \cdot {2^x}-16 \cdot {3^x} < 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{3^4} \cdot {2^x} < {2^4} \cdot {3^x}\left| {:\left( {{3^4} \cdot {3^x} > 0} \right)\,\,\,\,\,\,\, \Leftrightarrow } \right.\)
\( \Leftrightarrow \,\,\,\,\,\,\,\dfrac{{{2^x}}}{{{3^x}}} < \dfrac{{{2^4}}}{{{3^4}}}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\left( {\dfrac{2}{3}} \right)^x} < {\left( {\dfrac{2}{3}} \right)^4}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x > 4\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x\, \in \,\left( {4;\infty } \right).\)
Следовательно, наименьшее целое решение \(x = 5.\)
Ответ: 5.