Hướng dẫn Giải Bài 7 (Trang 90, SGK Toán Giải Tích 12)

Hướng dẫn giải Bài 7 (Trang 90 SGK Toán Giải Tích 12)

Bài 7 (Trang 90 SGK Toán Giải Tích 12): Giải các phương trình: a) <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>3</mn><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></msup><mo>+</mo><mn>3</mn><mo>.</mo><msup><mn>5</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></msup><mo>=</mo><msup><mn>5</mn><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></msup><mo>+</mo><msup><mn>3</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></msup></math>; b) <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>25</mn><mi>x</mi></msup><mo>-</mo><mn>6</mn><mo>.</mo><msup><mn>5</mn><mi>x</mi></msup><mo>+</mo><mn>5</mn><mo>=</mo><mn>0</mn></math>; c) <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><msup><mn>9</mn><mi>x</mi></msup><mo>+</mo><msup><mn>12</mn><mi>x</mi></msup><mo>-</mo><mn>3</mn><mo>.</mo><msup><mn>16</mn><mi>x</mi></msup><mo>=</mo><mn>0</mn></math>; d) <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>7</mn></msub><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>.</mo><msub><mi>log</mi><mn>7</mn></msub><mi>x</mi><mo>=</mo><msub><mi>log</mi><mn>7</mn></msub><mi>x</mi></math>; e) <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>3</mn></msub><mi>x</mi><mo>+</mo><msub><mi>log</mi><msqrt><mn>3</mn></msqrt></msub><mi>x</mi><mo>+</mo><msub><mi>log</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msub><mi>x</mi><mo>=</mo><mn>6</mn></math>; g) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfrac><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mi>log</mi><mi>x</mi></math> Hướng dẫn giải: a) <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>3</mn><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></msup><mo> </mo><mo>+</mo><mo> </mo><mn>3</mn><mo>.</mo><mo> </mo><msup><mn>5</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></msup><mo> </mo><mo>=</mo><mo> </mo><msup><mn>5</mn><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></msup><mo> </mo><mo>+</mo><mo> </mo><mn>3</mn><mo>.</mo><mo> </mo><msup><mn>3</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></msup><mo> </mo><mo>⇔</mo><mo> </mo><mn>81</mn><mo>.</mo><msup><mn>3</mn><mi>x</mi></msup><mo> </mo><mo>+</mo><mo> </mo><mn>375</mn><mo>.</mo><msup><mn>3</mn><mn>3</mn></msup><mo> </mo><mo>=</mo><mo> </mo><mn>625</mn><mo>.</mo><msup><mn>5</mn><mi>x</mi></msup><mo> </mo><mo>+</mo><mo> </mo><mn>27</mn><mo>.</mo><mn>3</mn><mspace linebreak="newline"/><mspace linebreak="newline"/><mo>⇔</mo><mo> </mo><mn>81</mn><mo>.</mo><msup><mn>3</mn><mi>x</mi></msup><mo> </mo><mo>-</mo><mo> </mo><mn>27</mn><mo>.</mo><msup><mn>3</mn><mi>x</mi></msup><mo> </mo><mo>=</mo><mo> </mo><mn>625</mn><mo>.</mo><msup><mn>5</mn><mi>x</mi></msup><mo> </mo><mo>-</mo><mo> </mo><mn>375</mn><mo>.</mo><msup><mn>5</mn><mi>x</mi></msup><mo> </mo><mo>⇔</mo><mo> </mo><mn>54</mn><mo>.</mo><msup><mn>3</mn><mi>x</mi></msup><mo> </mo><mo>=</mo><mo> </mo><mn>250</mn><mo>.</mo><msup><mn>5</mn><mi>x</mi></msup><mspace linebreak="newline"/><mspace linebreak="newline"/><mo>⇔</mo><mo> </mo><msup><mfenced><mfrac><mn>3</mn><mn>5</mn></mfrac></mfenced><mi>x</mi></msup><mo> </mo><mo>=</mo><mo> </mo><msup><mfenced><mfrac><mn>5</mn><mn>3</mn></mfrac></mfenced><mn>3</mn></msup><mo> </mo><mo>⇔</mo><mo> </mo><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>3</mn><mo>.</mo></math> Vậy S = {-3} b) Đặt t = 5x (t > 0 ), ta có phương trình: <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">t</mi><mn>2</mn></msup><mo> </mo><mo>-</mo><mo> </mo><mn>6</mn><mi mathvariant="normal">t</mi><mo> </mo><mo>+</mo><mo> </mo><mn>5</mn><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn><mo> </mo><mo>⇔</mo><mo> </mo><mo>[</mo><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">t</mi><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">t</mi><mo> </mo><mo>=</mo><mo> </mo><mn>5</mn></mtd></mtr></mtable><mo> </mo><mo>⇔</mo><mo> </mo><mo>[</mo><mtable columnalign="left"><mtr><mtd><msup><mn>5</mn><mi mathvariant="normal">x</mi></msup><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn></mtd></mtr><mtr><mtd><msup><mn>5</mn><mi mathvariant="normal">x</mi></msup><mo> </mo><mo>=</mo><mo> </mo><mn>5</mn></mtd></mtr></mtable><mo> </mo><mo>⇔</mo><mo> </mo><mo>[</mo><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn></mtd></mtr></mtable></math> Vậy S = {0;1} c) Chia hai vế phương trình cho 9x (9x > 0) ta được: <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>+</mo><msup><mfenced><mfrac><mn>4</mn><mn>3</mn></mfrac></mfenced><mi>x</mi></msup><mo>-</mo><mn>3</mn><mo>.</mo><msup><mfenced><mfrac><mn>16</mn><mn>9</mn></mfrac></mfenced><mi>x</mi></msup><mo>=</mo><mn>0</mn><mo> </mo><mo>.</mo><mo> </mo><mo> </mo><mo> </mo><mi>Đặt</mi><mo> </mo><mi mathvariant="normal">t</mi><mo>=</mo><msup><mfenced><mfrac><mn>4</mn><mn>3</mn></mfrac></mfenced><mi mathvariant="normal">x</mi></msup><mo> </mo><mo>(</mo><mi mathvariant="normal">t</mi><mo>></mo><mn>0</mn><mo>)</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Ta</mi><mo> </mo><mi>có</mi><mo>:</mo><mo> </mo><mn>4</mn><mo>+</mo><mi mathvariant="normal">t</mi><mo>-</mo><mn>3</mn><msup><mi mathvariant="normal">t</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo> </mo><mo>⇔</mo><mo> </mo><mo>[</mo><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">t</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>1</mn><mo> </mo><mo>(</mo><mi>loại</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">t</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mtd></mtr></mtable></math> Với <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">t</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mo> </mo><mo>⇔</mo><msup><mfenced><mfrac><mn>4</mn><mn>3</mn></mfrac></mfenced><mi mathvariant="normal">x</mi></msup><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mo> </mo><mo>⇔</mo><mo> </mo><mi mathvariant="normal">x</mi><mo>=</mo><mn>1</mn></math> Vậy S = {1}. d) Điều kiện: x>1, khi đó log7x > 0. <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>7</mn></msub><mfenced><mrow><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>.</mo><msub><mi>log</mi><mn>7</mn></msub><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><msub><mi>log</mi><mn>7</mn></msub><mi mathvariant="normal">x</mi><mo> </mo><mo>⇔</mo><mo> </mo><msub><mi>log</mi><mn>7</mn></msub><mfenced><mrow><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn><mo> </mo><mo>⇔</mo><mo> </mo><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>8</mn></math> Vậy S = {8} e) Điều kiện: x > 0 <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>3</mn></msub><mi>x</mi><mo> </mo><mo> </mo><mo>+</mo><mo> </mo><msub><mi>log</mi><msqrt><mn>3</mn></msqrt></msub><mi>x</mi><mo> </mo><mo>+</mo><mo> </mo><msub><mi>log</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msub><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>6</mn><mo> </mo><mo>⇔</mo><mo> </mo><msub><mi>log</mi><mn>3</mn></msub><mi>x</mi><mo> </mo><mo>+</mo><mo> </mo><mn>2</mn><msub><mi>log</mi><mn>3</mn></msub><mi>x</mi><mo> </mo><mo>-</mo><mo> </mo><msub><mi>log</mi><mn>3</mn></msub><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>6</mn><mo> </mo><mo>⇔</mo><mo> </mo><msub><mi>log</mi><mn>3</mn></msub><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>3</mn><mo> </mo><mo>⇔</mo><mo> </mo><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>27</mn></math> Vậy S = {27}. g) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfrac><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>8</mn></mrow><mrow><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mi>logx</mi><mo> </mo><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">x</mi><mo>></mo><mn>0</mn></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>8</mn></mrow><mrow><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mi mathvariant="normal">x</mi></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">x</mi><mo>></mo><mn>0</mn></mtd></mtr><mtr><mtd><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>8</mn><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mi mathvariant="normal">x</mi><mo>=</mo><mn>4</mn></math> Vậy S = {4}.