Cho phương trình \({\log _3}^2\left( {9x} \right)...

D

- Hướng dẫn giải

\(\begin{array}{*{20}{l}}{{{\log }_3}^2\left( {9x} \right) - \left( {m + 5} \right){{\log }_3}x + 3m - 10 = 0}\\{ \Rightarrow 4{{\log }_3}^2x - \left( {m + 5} \right){{\log }_3}x + 3m - 10 = 0}\\{t = {{\log }_3}x \Rightarrow 1 \le x = {3^t} \le 81 \Rightarrow 0 \le t \le 4}\\{PT \Leftrightarrow 4{t^2} - (m + 5)t + 3m - 10 = 0}\\{\Delta = {{\left( {m + 5} \right)}^2} - 16\left( {3m - 10} \right) > 0}\\{ \Leftrightarrow m > 19 + 4\sqrt {11} ;m < 19 - 4\sqrt {11} }\\{{t_1} + {t_2} = \frac{{m + 5}}{4} < 4 + 4 \Rightarrow m < 27}\\{0 < {t_1}.{t_2} = \frac{{3m - 10}}{4} < 4.4 \Rightarrow \frac{{10}}{3} < m < \frac{{74}}{3}}\\\begin{array}{l} \Rightarrow \frac{{10}}{3} < m < 19 - 4\sqrt {11} \\ \Rightarrow m \in \left\{ {4;\,\,5} \right\}\end{array}\end{array}\)