ОДЗ: \(\left\{ {\begin{array}{*{20}{c}}{x-2 > 0,}\\{x + 2 > 0,}\\{2x-1 > 0\,}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x > 2,\,\,\,}\\{x > -2,}\\{x > 0,5}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x \in \left( {2;\infty } \right).\)
\({\log _3}\left( {x-2} \right) + {\log _3}\left( {x + 2} \right) = {\log _3}\left( {2x-1} \right)\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\log _3}\left( {x-2} \right)\left( {x + 2} \right) = {\log _3}\left( {2x-1} \right)\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\)
\( \Leftrightarrow \,\,\,\,\,\,\,{x^2}-4 = 2x-1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{x^2}-2x-3 = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = -1,}\\{x = 3.\,\,\,}\end{array}} \right.\)
Корень \(x = -1\) не удовлетворяет ОДЗ.
Ответ: 3.