\(\left( {{{\log }_7}9-{{\log }_6}9} \right) \cdot {\log _3}\left( {x-13} \right) \ge 0.\)
Так как \({\log _7}9 = \dfrac{1}{{{{\log }_9}7}}\) и \({\log _6}9 = \dfrac{1}{{{{\log }_9}6}}\), при этом:
\({\log _9}7 > {\log _9}6 > 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\dfrac{1}{{{{\log }_9}7}} < \dfrac{1}{{{{\log }_9}6}}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\log _7}9 < {\log _6}9\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\log _7}9-{\log _6}9 < 0,\)
то:
\(\left( {{{\log }_7}9-{{\log }_6}9} \right) \cdot {\log _3}\left( {x-13} \right) \ge 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\log _3}\left( {x-13} \right) \le 0\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,{\log _3}\left( {x-13} \right) \le {\log _3}1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,0 < x-13 \le 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,13 < x \le 14\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,x\, \in \,\,\left( {13;14} \right].\)
Целое решение 14. Их количество равно 1.
Ответ: 1.