Bài 5: Phương trình mũ và phương trình lôgarit
Hướng dẫn giải Bài 2 (Trang 84 SGK Toán Giải Tích 12)
<p><strong>Bài 2 (Trang 84 SGK Toán Giải Tích 12):</strong></p>
<p>Giải các phương trình mũ :</p>
<p>a) <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>3</mn><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>1</mn></mrow></msup><mo> </mo><mo>+</mo><mo> </mo><msup><mn>3</mn><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo> </mo><mo>=</mo><mo> </mo><mn>108</mn></math>;</p>
<p>b) <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mo> </mo><msup><mn>2</mn><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></msup><mo> </mo><mo>+</mo><msup><mn>2</mn><mi>x</mi></msup><mo> </mo><mo>=</mo><mo> </mo><mn>28</mn></math>;</p>
<p>c) <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>64</mn><mi>x</mi></msup><mo> </mo><mo>-</mo><mo> </mo><msup><mn>8</mn><mi>x</mi></msup><mo> </mo><mo>-</mo><mn>56</mn><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn></math>;</p>
<p>d) <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><msup><mn>4</mn><mi>x</mi></msup><mo> </mo><mo>-</mo><mo> </mo><mn>2</mn><mo>.</mo><msup><mn>6</mn><mi>x</mi></msup><mo> </mo><mo>=</mo><mo> </mo><msup><mn>9</mn><mi>x</mi></msup></math></p>
<p><strong><em>Hướng dẫn giải:</em></strong></p>
<p>a) <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>3</mn><mrow><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn></mrow></msup><mo> </mo><mo>+</mo><mo> </mo><msup><mn>3</mn><mrow><mn>2</mn><mi mathvariant="normal">x</mi></mrow></msup><mo> </mo><mo>=</mo><mo> </mo><mn>108</mn><mo>⇔</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>.</mo><msup><mn>3</mn><mrow><mn>2</mn><mi mathvariant="normal">x</mi></mrow></msup><mo> </mo><mo>+</mo><mo> </mo><msup><mn>3</mn><mrow><mn>2</mn><mi mathvariant="normal">x</mi></mrow></msup><mo> </mo><mo>=</mo><mo> </mo><mn>108</mn><mo>⇔</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>.</mo><msup><mn>3</mn><mrow><mn>2</mn><mi mathvariant="normal">x</mi></mrow></msup><mo> </mo><mo>=</mo><mo> </mo><mn>108</mn><mo> </mo><mo>⇔</mo><msup><mn>3</mn><mrow><mn>2</mn><mi mathvariant="normal">x</mi></mrow></msup><mo> </mo><mo>=</mo><mo> </mo><mn>81</mn><mo> </mo><mo>⇔</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>4</mn><mo>⇔</mo><mo> </mo><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn></math>. Vậy S = {2}.</p>
<p>b) <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></msup><mo> </mo><mo>+</mo><mo> </mo><msup><mn>2</mn><mrow><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn></mrow></msup><mo> </mo><mo>+</mo><msup><mn>2</mn><mi mathvariant="normal">x</mi></msup><mo> </mo><mo>=</mo><mn>28</mn><mo>⇔</mo><msup><mn>2</mn><mi mathvariant="normal">x</mi></msup><mfenced><mrow><mn>2</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mn>28</mn><mo>⇔</mo><msup><mn>2</mn><mi mathvariant="normal">x</mi></msup><mo> </mo><mo>=</mo><mo> </mo><mn>8</mn><mo>⇔</mo><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>3</mn></math>. Vậy S = {3}.</p>
<p>c) Đặt t = 8<sup>x</sup> (t > 0), ta có phương trình: t<sup>2</sup> - t - 56 = 0 => t = 8 hoặc t = -7 (loại).</p>
<p>Với t = 9 => 8<sup>x</sup> = 8 => x = 1. Vậy S = {1}.</p>
<p>d) Chia hai vế của phương trình cho 9<sup>x</sup> (9<sup>x</sup> > 0) ta được: <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mo> </mo><msup><mfenced><mfrac><mn>4</mn><mn>9</mn></mfrac></mfenced><mi>x</mi></msup><mo>-</mo><mo> </mo><mn>2</mn><msup><mfenced><mfrac><mn>2</mn><mn>3</mn></mfrac></mfenced><mi>x</mi></msup><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn></math></p>
<p>Đặt <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">t</mi><mo> </mo><mo>=</mo><mo> </mo><msup><mfenced><mfrac><mn>2</mn><mn>3</mn></mfrac></mfenced><mi mathvariant="normal">x</mi></msup><mo> </mo><mfenced><mrow><mi mathvariant="normal">t</mi><mo> </mo><mo>></mo><mn>0</mn></mrow></mfenced></math> ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mi mathvariant="normal">t</mi><mn>2</mn></msup><mo> </mo><mo>-</mo><mo> </mo><mn>2</mn><mi mathvariant="normal">t</mi><mo> </mo><mo>-</mo><mo> </mo><mn>1</mn><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn><mo>⇔</mo><mrow><mo> </mo><mo>[</mo><mtable columnspacing="2px" columnalign="right center left"><mtr><mtd><mi mathvariant="normal">t</mi></mtd><mtd><mo>=</mo></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">t</mi></mtd><mtd><mo>=</mo></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mtd></mtr></mtable></mrow></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">t</mi><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn><mo>⇔</mo><msup><mfenced><mfrac><mn>2</mn><mn>3</mn></mfrac></mfenced><mi mathvariant="normal">x</mi></msup><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn><mo>⇔</mo><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn></math>. Vậy S = {0}.</p>
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