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Giải mục 4 trang 31, 32 SGK Toán 11 tập 2 - Chân trời sáng tạo

Biết rằng máu của người bình thường có độ pH từ 7,30 đến 7,45

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Hoạt động 6

Biết rằng máu của người bình thường có độ pH từ 7,30 đến 7,45 (nguồn: Hoá học 11, NXB Giáo dục Việt Nam, năm 2017, trang 15). Nồng độ H+ trong máu nhận giá trị trong miền nào?

Phương pháp giải:

Xét tính đồng biến, nghịch biến của hàm số \(pH =  - \log x\).

Lời giải chi tiết:

\(pH =  - \log x = {\log _{{{10}^{ - 1}}}}x = {\log _{\frac{1}{{10}}}}x\) Do \(0 < \frac{1}{{10}} < 1\) nên hàm số \(pH = {\log _{\frac{1}{{10}}}}x\) nghịch biến trên \(\left( {0; + \infty } \right)\). Ta có: \(\begin{array}{l}pH = 7,3 \Leftrightarrow 7,3 = {\log _{\frac{1}{{10}}}}x \Leftrightarrow x = {\left( {\frac{1}{{10}}} \right)^{7,3}} \approx 5,{01.10^{ - 8}}\\pH = 7,45 \Leftrightarrow 7,45 = {\log _{\frac{1}{{10}}}}x \Leftrightarrow x = {\left( {\frac{1}{{10}}} \right)^{7,45}} \approx 3,{55.10^{ - 8}}\end{array}\)

Vì hàm số nghịch biến trên \(\left( {0; + \infty } \right)\) nên nồng độ H+🌼 trong máu nhận giá trị trong miền từ \(3,{55.10^{ - 8}}\) đến \(5,{01.10^{ - 8}}\).

Thực hành 4

Giải các bất phương trình sau: a) \({\log _{\frac{1}{3}}}\left( {x + 1} \right) < 2\);    b) \({\log _5}\left( {x + 2} \right) \le 1\).

Phương pháp giải:

Bước 1: Tìm ĐKXĐ.

Bước 2:🥂 Đưa 2 vế của phương trình về cùng cơ số và giải phương trình.

Bước 3: Kết luận.

Lời giải chi tiết:

a) \({\log _{\frac{1}{3}}}\left( {x + 1} \right) < 2\) Điều kiện: \(x + 1 > 0 \Leftrightarrow x >  - 1\) \({\log _{\frac{1}{3}}}(x + 1) < 2 \Leftrightarrow x + 1 > {\left( {\frac{1}{3}} \right)^2} \Leftrightarrow x > \frac{{ - 8}}{9}\). Vậy nghiệm của bất phương trình là \(x >  - \frac{8}{9}\). b) \({\log _5}\left( {x + 2} \right) \le 1\) Điều kiện: \(x + 2 > 0 \Leftrightarrow x >  - 2\) \(BPT \Leftrightarrow x + 2 \le {5^1} \Leftrightarrow x + 2 \le 5 \Leftrightarrow x \le 3\) Kết hợp với điều kiện ta được nghiệm của bất phương trình là \( - 2 < x \le 3\).

Vận dụng 2

Nước uống đạt tiêu chuẩn phải có độ pH nằm trong khoảng từ 6,5 đến 8,5 (theo Quy chuẩn Việt Nam QCVN 01:2009/BYT). Nồng độ H+🥃 trong nước uống tiêu chuẩn phải nằm trong khoảng nào?

Phương pháp giải:

Giải bất phương trình \(6,5 \le pH \le 8,5\).

Lời giải chi tiết:

Ta có: \(\begin{array}{l}6,5 \le pH \le 8,5 \Leftrightarrow 6,5 \le  - \log x \le 8,5 \Leftrightarrow  \Leftrightarrow  - 6,5 \ge \log x \ge  - 8,5\\ \Leftrightarrow {10^{ - 6,5}} \ge x \ge {10^{ - 8,5}} \Leftrightarrow 3,{16.10^{ - 7}} \ge x \ge 3,{16.10^{ - 9}}\end{array}\)

Vậy nồng độ H+🥀 trong nước uống tiêu chuẩn phải nằm trong khoảng từ \(3,{16.10^{ - 9}}\) đến \(3,{16.10^{ - 7}}\).

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