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Giải bài 6 trang 22 sách bài tập toán 11 - Chân trời sáng tạo tập 2

Tìm tất cả các số nguyên x thỏa mãn ({log _3}left( {x - 2} right).{log _3}left( {x - 1} right) < 0).

♈Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Chân trời sáng tạo

Toán - Văn - Anh - Lí - Hóa - Sinh
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Đề bài

Tìm tất cả các số nguyên x thỏa mãn \({\log _3}\left( {x - 2} \right).{\log _3}\left( {x - 1} \right) < 0\).

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về bất phương trình lôgarit để giải bất phương trình Bảng tổng kết về nghiệm của các bất phương trình:
Bất phương trình \(a > 1\) \(0 < a < 1\)
\({\log _a}x > b\) \(x > {a^b}\) \(0 < x < {a^b}\)
\({\log _a}x \ge b\) \(x \ge {a^b}\) \(0 < x \le {a^b}\)
\({\log _a}x < b\) \(0 < x < {a^b}\) \(x > {a^b}\)
\({\log _a}x \le b\) \(0 < x \le {a^b}\) \(x \ge {a^b}\)

Lời giải chi tiết

Điều kiện: \(x > 2\) \({\log _3}\left( {x - 2} \right).{\log _3}\left( {x - 1} \right) < 0 \) \( \Leftrightarrow \left[ \begin{array}{l}\left\{ \begin{array}{l}{\log _3}\left( {x - 2} \right) < 0\\{\log _3}\left( {x - 1} \right) > 0\end{array} \right.\\\left\{ \begin{array}{l}{\log _3}\left( {x - 2} \right) > 0\\{\log _3}\left( {x - 1} \right) < 0\end{array} \right.\end{array} \right. \) \( \Leftrightarrow \left[ \begin{array}{l}\left\{ \begin{array}{l}x - 2 < 1\\x - 1 > 1\end{array} \right.\\\left\{ \begin{array}{l}x - 2 > 1\\x - 1 < 1\end{array} \right.\end{array} \right. \) \( \Leftrightarrow \left[ \begin{array}{l}\left\{ \begin{array}{l}x < 3\\x > 2\end{array} \right.\\\left\{ \begin{array}{l}x > 3\\x < 2\end{array} \right.\left( {VL} \right)\end{array} \right. \) \( \Leftrightarrow 2 < x < 3\) Mà x là số nguyên nên không có giá trị nào của x thỏa mãn bài toán.

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