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Giải bài 31 trang 39 sách bài tập toán 11 - Cánh diều

Cho \(x > 0,y > 0\) thỏa mãn \({x^2} + 4{y^2} = 6xy.\) Chứng minh rằng:

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

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Đề bài

Cho \(x > 0,y > 0\) thỏa mãn \({x^2} + 4{y^2} = 6xy.\) Chứng minh rằng: \(2\log \left( {x + 2y} \right) = 1 + \log x + \log y.\)

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 chứng minh.

Lời giải chi tiết

Theo đề bài: \({x^2} + 4{y^2} = 6xy \Leftrightarrow {x^2} + 4xy + 4{y^2} = 10xy \Leftrightarrow {\left( {x + 2y} \right)^2} = 10xy.\)\(\begin{array}{l} \Rightarrow 2\log \left( {x + 2y} \right) = \log {\left( {x + 2y} \right)^2} = \log \left( {10xy} \right) = \log 10 + \log xy\\ = 1 + \log x + \log y.\end{array}\)

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