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Giải bài 26 trang 38 sách bài tập toán 11 - Cánh diều

Cho \(a > 0,b > 0\) thỏa mãn \({a^2} + {b^2} = 7ab\). Khi đó, \(\log \left( {a + b} \right)\) bằng :

🔜Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Cánh diều

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

Cho \(a > 0,b > 0\) thỏa mãn \({a^2} + {b^2} = 7ab\). Khi đó,  \(\log \left( {a + b} \right)\) bằng : A. \(\log 9 + \frac{1}{2}\left( {\log a + \log b} \right).\) B. \(\log 3 + \frac{1}{2}\log a.\log b.\) C. \(\log 3 + \frac{1}{2}\log a + \log b.\) D. \(\log 3 + \frac{1}{2}\left( {\log a + \log b} \right).\)

Phương pháp giải - Xem chi tiết

Sử dụng các tính chất của logarit và hằng đẳng thức  \({\left( {m + n} \right)^2} = {m^2} + 2mn + {n^2}\) để tính giá trị biểu thức.

Lời giải chi tiết

Theo đề bài: \({a^2} + {b^2} = 7ab \Leftrightarrow {a^2} + 2ab + {b^2} = 9ab \Leftrightarrow {\left( {a + b} \right)^2} = 9ab.\)\(\begin{array}{l} \Rightarrow \log \left( {a + b} \right) = \frac{1}{2}\log {\left( {a + b} \right)^2} = \frac{1}{2}\log \left( {9ab} \right) = \frac{1}{2}\log {3^2} + \frac{1}{2}\log ab\\ = \log 3 + \frac{1}{2}\left( {\log a + \log b} \right).\end{array}\)Đáp án D.

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