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Câu hỏi: Có bao nhiêu giá trị nguyên $y$ sao cho tồn tại số thực $x$ thỏa mãn
B. $9$.
C. $11$.
D. $10$.
${{\log }_{4}}\left( \sqrt{{{x}^{2}}+{{3}^{y}}}-x \right).{{\log }_{3}}\left( \sqrt{{{x}^{2}}+{{3}^{y}}}+x \right)={{y}^{2}}-7y$ ?
A. $8$.B. $9$.
C. $11$.
D. $10$.
Nhận xét: $\left( \sqrt{{{x}^{2}}+{{3}^{y}}}-x \right)\left( \sqrt{{{x}^{2}}+{{3}^{y}}}+x \right)={{3}^{y}}$
Suy ra ${{\log }_{3}}\left( \sqrt{{{x}^{2}}+{{3}^{y}}}-x \right)+{{\log }_{3}}\left( \sqrt{{{x}^{2}}+{{3}^{y}}}+x \right)=y$ (1)
Phương trình đã cho ${{\log }_{4}}3.{{\log }_{3}}\left( \sqrt{{{x}^{2}}+{{3}^{y}}}-x \right).{{\log }_{3}}\left( \sqrt{{{x}^{2}}+{{3}^{y}}}+x \right)={{y}^{2}}-7y$
Suy ra ${{\log }_{3}}\left( \sqrt{{{x}^{2}}+{{3}^{y}}}-x \right).{{\log }_{3}}\left( \sqrt{{{x}^{2}}+{{3}^{y}}}+x \right)=\left( {{y}^{2}}-7y \right).{{\log }_{3}}4$ (2)
$\left( 1 \right),\left( 2 \right)$ theo yêu cầu bài toán ta cần ${{y}^{2}}-4.{{\log }_{3}}4\left( {{y}^{2}}-7y \right)\ge 0$
$\Leftrightarrow \left( 1-4{{\log }_{3}}4 \right).{{y}^{2}}+\left( 28{{\log }_{3}}4 \right)y\ge 0$ $\Leftrightarrow 0\le y\le \underbrace{\dfrac{28{{\log }_{3}}4}{4{{\log }_{3}}4-1}}_{\approx 8,73}$.
Do $y\in \mathbb{Z}\Rightarrow y\in \left\{ 0;1;2;...;8 \right\}$.
Suy ra ${{\log }_{3}}\left( \sqrt{{{x}^{2}}+{{3}^{y}}}-x \right)+{{\log }_{3}}\left( \sqrt{{{x}^{2}}+{{3}^{y}}}+x \right)=y$ (1)
Phương trình đã cho ${{\log }_{4}}3.{{\log }_{3}}\left( \sqrt{{{x}^{2}}+{{3}^{y}}}-x \right).{{\log }_{3}}\left( \sqrt{{{x}^{2}}+{{3}^{y}}}+x \right)={{y}^{2}}-7y$
Suy ra ${{\log }_{3}}\left( \sqrt{{{x}^{2}}+{{3}^{y}}}-x \right).{{\log }_{3}}\left( \sqrt{{{x}^{2}}+{{3}^{y}}}+x \right)=\left( {{y}^{2}}-7y \right).{{\log }_{3}}4$ (2)
$\left( 1 \right),\left( 2 \right)$ theo yêu cầu bài toán ta cần ${{y}^{2}}-4.{{\log }_{3}}4\left( {{y}^{2}}-7y \right)\ge 0$
$\Leftrightarrow \left( 1-4{{\log }_{3}}4 \right).{{y}^{2}}+\left( 28{{\log }_{3}}4 \right)y\ge 0$ $\Leftrightarrow 0\le y\le \underbrace{\dfrac{28{{\log }_{3}}4}{4{{\log }_{3}}4-1}}_{\approx 8,73}$.
Do $y\in \mathbb{Z}\Rightarrow y\in \left\{ 0;1;2;...;8 \right\}$.
Đáp án B.