Hướng dẫn Giải Bài 5 (Trang 68, SGK Toán Giải Tích 12)

Hướng dẫn giải Bài 5 (Trang 68 SGK Toán Giải Tích 12)

Bài 5 (Trang 68 SGK Toán Giải Tích 12): a) Cho a = log30(3), b = log30(5). Hãy tính log30(1350) theo a, b; b) Cho c = log15(3). Hãy tính log25(15) theo c. Hướng dẫn giải: a) Ta có 1350 = 32.5.30 Do đó log30(1350) = 2.log30(3) + log30(5) + log30(30) = 2a + b + 1. b) Áp dụng công thức đổi cơ số ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>25</mn></msub><mfenced><mn>15</mn></mfenced><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><msub><mi>log</mi><mn>3</mn></msub><mfenced><mn>15</mn></mfenced></mrow><mrow><msub><mi>log</mi><mn>3</mn></msub><mfenced><mn>25</mn></mfenced></mrow></mfrac><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><msub><mi>log</mi><mn>3</mn></msub><mfenced><mrow><mn>3</mn><mo>.</mo><mn>5</mn></mrow></mfenced></mrow><mrow><msub><mi>log</mi><mn>3</mn></msub><mfenced><msup><mn>5</mn><mn>2</mn></msup></mfenced></mrow></mfrac><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><mn>1</mn><mo>+</mo><msub><mi>log</mi><mn>3</mn></msub><mfenced><mn>5</mn></mfenced></mrow><mrow><mn>2</mn><msub><mi>log</mi><mn>3</mn></msub><mfenced><mn>5</mn></mfenced></mrow></mfrac></math> Do đó ta phải tìm log3(5). Theo đề bài: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">c</mi><mo> </mo><mo>=</mo><mo> </mo><msub><mi>log</mi><mn>15</mn></msub><mfenced><mn>3</mn></mfenced><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><msub><mi>log</mi><mn>3</mn></msub><mfenced><mn>3</mn></mfenced></mrow><mrow><msub><mi>log</mi><mn>3</mn></msub><mfenced><mn>15</mn></mfenced></mrow></mfrac><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msub><mi>log</mi><mn>3</mn></msub><mfenced><mn>5</mn></mfenced></mrow></mfrac><mo> </mo><mo>⇒</mo><mo> </mo><msub><mi>log</mi><mn>3</mn></msub><mfenced><mn>5</mn></mfenced><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>1</mn><mi mathvariant="normal">c</mi></mfrac><mo>-</mo><mn>1</mn></math> Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>25</mn></msub><mfenced><mn>15</mn></mfenced><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><mn>1</mn><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi mathvariant="normal">c</mi></mfrac></mstyle><mo>-</mo><mn>1</mn></mrow><mrow><mn>2</mn><mo>(</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi mathvariant="normal">c</mi></mfrac></mstyle><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mfrac><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>1</mn><mrow><mn>2</mn><mo>(</mo><mn>1</mn><mo>-</mo><mi mathvariant="normal">c</mi><mo>)</mo></mrow></mfrac></math>