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Câu hỏi: Tổng tất cả các nghiệm của phương trình ${{\log }_{3}}\left( {{9}^{x}}-{{5.3}^{x}}+7 \right)=x+1$ bằng
A. ${{\log }_{7}}3$.
B. $1+{{\log }_{3}}7$.
C. ${{\log }_{3}}7$.
D. $1+{{\log }_{3}}7$.
A. ${{\log }_{7}}3$.
B. $1+{{\log }_{3}}7$.
C. ${{\log }_{3}}7$.
D. $1+{{\log }_{3}}7$.
TXD: $D=\mathbb{R}$
$\begin{aligned}
& {{\log }_{3}}\left( {{9}^{x}}-{{5.3}^{x}}+7 \right)=x+1 \\
& \Leftrightarrow {{9}^{x}}-{{5.3}^{x}}+7={{3}^{x+1}} \\
& {{\left( {{3}^{x}} \right)}^{2}}-{{5.3}^{x}}+7-{{3.3}^{x}}=0 \\
& {{\left( {{3}^{x}} \right)}^{2}}-{{8.3}^{x}}+7=0 \\
\end{aligned}$
Đặt $t={{3}^{x}},t>0$
$\begin{aligned}
& {{t}_{1}}.{{t}_{2}}={{3}^{{{x}_{1}}+{{x}_{2}}}} \\
& \Leftrightarrow 7={{3}^{{{x}_{1}}+{{x}_{2}}}}\Rightarrow {{x}_{1}}+{{x}_{2}}={{\log }_{3}}7 \\
\end{aligned}$
$\begin{aligned}
& {{\log }_{3}}\left( {{9}^{x}}-{{5.3}^{x}}+7 \right)=x+1 \\
& \Leftrightarrow {{9}^{x}}-{{5.3}^{x}}+7={{3}^{x+1}} \\
& {{\left( {{3}^{x}} \right)}^{2}}-{{5.3}^{x}}+7-{{3.3}^{x}}=0 \\
& {{\left( {{3}^{x}} \right)}^{2}}-{{8.3}^{x}}+7=0 \\
\end{aligned}$
Đặt $t={{3}^{x}},t>0$
$\begin{aligned}
& {{t}_{1}}.{{t}_{2}}={{3}^{{{x}_{1}}+{{x}_{2}}}} \\
& \Leftrightarrow 7={{3}^{{{x}_{1}}+{{x}_{2}}}}\Rightarrow {{x}_{1}}+{{x}_{2}}={{\log }_{3}}7 \\
\end{aligned}$
Đáp án C.