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Giải bài 6.25 trang 14 sách bài tập toán 11 - Kết nối tri thức với cuộc sống

Cho hàm số lôgarit \(f\left( x \right) = {\rm{lo}}{{\rm{g}}_a}x\,\,\,\,(0 < a \ne 1)\). Chứng minh rằng:

Tổng hợ𓄧p đề thi học kì 2 lớp 11 tất cả các môn - Kết nối✤ tri thức

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

Cho hàm số lôgarit \(f\left( x \right) = {\rm{lo}}{{\rm{g}}_a}x\,\,\,\,(0 < a \ne 1)\). Chứng minh rằng: a) \(f\left( {\frac{1}{x}} \right) =  - f\left( x \right)\) b) \(f\left( {{x^\alpha }} \right) = \alpha f\left( x \right)\)

Phương pháp giải - Xem chi tiết

Áp dụng quy tắc tính lôgarit Giả sử a là số thực dương khác \(1,\,M\) và \(N\) là các số thực dương, \(\alpha \) là số thực tuỳ ý. \(\begin{array}{l}{\log _a}(MN) = {\log _a}M + {\log _a}N;\\{\log _a}\left( {\frac{M}{N}} \right) = {\log _a}M - {\log _a}N;{\log _a}\frac{1}{b} = {\log _a}1 - {\log _a}b = {\log _a}b\\{\log _a}{M^a} = \alpha {\log _a}M.\end{array}\)

Lời giải chi tiết

a) \(f\left( {\frac{1}{x}} \right) = {\rm{lo}}{{\rm{g}}_a}\frac{1}{x} =  - {\rm{lo}}{{\rm{g}}_a}x =  - f\left( x \right)\)b) \(f\left( {{x^\alpha }} \right) = {\rm{lo}}{{\rm{g}}_a}{x^\alpha } = \alpha {\rm{lo}}{{\rm{g}}_a}x = \alpha f\left( x \right)\).

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