Bài 5: Phương trình mũ và phương trình lôgarit
Hướng dẫn giải Bài 3 (Trang 84 SGK Toán Giải Tích 12)
<p><strong>Bài 3 (Trang 84 SGK Toán Giải Tích 12):</strong></p>
<p>Giải các phương trình lôgarit :</p>
<p>a) <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>3</mn></msub><mfenced><mrow><mn>5</mn><mi>x</mi><mo> </mo><mo>+</mo><mo> </mo><mn>3</mn></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><msub><mi>log</mi><mn>3</mn></msub><mfenced><mrow><mn>7</mn><mi>x</mi><mo> </mo><mo>+</mo><mo> </mo><mn>5</mn></mrow></mfenced></math>;</p>
<p>b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mo> </mo><mi>log</mi><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>11</mn></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mi>log</mi><mn>2</mn></math>;</p>
<p>c) <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>2</mn></msub><mfenced><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mo> </mo><mo>+</mo><mo> </mo><msub><mi>log</mi><mn>2</mn></msub><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mn>3</mn></math>;</p>
<p>d) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mi>log</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced></math>.</p>
<p><strong><em>Hướng dẫn giải:</em></strong></p>
<p>a) Điều kiện <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>5</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>3</mn><mo> </mo><mo>></mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>7</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>5</mn><mo> </mo><mo>></mo><mn>0</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mi mathvariant="normal">x</mi><mo>></mo><mo>-</mo><mfrac><mn>3</mn><mn>5</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>3</mn></msub><mfenced><mrow><mn>5</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><msub><mi>log</mi><mn>3</mn></msub><mfenced><mrow><mn>7</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>5</mn></mrow></mfenced><mo>⇔</mo><mn>5</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>3</mn><mo> </mo><mo>=</mo><mo> </mo><mn>7</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>5</mn><mo>⇔</mo><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>1</mn><mo> </mo><mo>(</mo><mi>loại</mi><mo>)</mo></math></p>
<p>Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo> </mo><mo>=</mo><mo> </mo><menclose notation="updiagonalstrike"><mi>O</mi></menclose></math></p>
<p>b) Điều kiện <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn><mo>></mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>11</mn><mo>></mo><mn>0</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mi mathvariant="normal">x</mi><mo>></mo><mfrac><mn>11</mn><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mo> </mo><mi>log</mi><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>11</mn></mrow></mfenced><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><mi>log</mi><mn>2</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇔</mo><mi>log</mi><mfrac><mrow><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>11</mn></mrow></mfrac><mo> </mo><mo>=</mo><mo> </mo><mi>log</mi><mn>2</mn><mo>⇔</mo><mfrac><mrow><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>11</mn></mrow></mfrac><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn><mo>⇔</mo><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn><mo> </mo><mo>=</mo><mo> </mo><mn>4</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>22</mn><mo>⇔</mo><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>7</mn><mo> </mo><mo>(</mo><mi>nhận</mi><mo>)</mo></math></p>
<p>Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo> </mo><mo>=</mo><mo> </mo><mfenced open="{" close=""><mfenced open="" close="}"><mn>7</mn></mfenced></mfenced></math></p>
<p>c) Điều kiện: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>5</mn></math></p>
<p>Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>2</mn></msub><mfenced><mrow><mi mathvariant="normal">x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mo>+</mo><msub><mi>log</mi><mn>2</mn></msub><mfenced><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mn>3</mn><mo>⇔</mo><msub><mi>log</mi><mn>2</mn></msub><mfenced><mrow><mi mathvariant="normal">x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><msub><mi>log</mi><mn>2</mn></msub><mn>8</mn><mo> </mo><mo>⇔</mo><mo> </mo><mfenced><mrow><mi mathvariant="normal">x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mn>8</mn><mo>⇔</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo> </mo><mo>-</mo><mo> </mo><mn>3</mn><mi mathvariant="normal">x</mi><mo>-</mo><mo> </mo><mn>18</mn><mo>=</mo><mn>0</mn><mo> </mo><mspace linebreak="newline"/><mo>⇔</mo><mo>[</mo><mtable columnspacing="2px" columnalign="right center left"><mtr><mtd><mi mathvariant="normal">x</mi></mtd><mtd><mo>=</mo></mtd><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">x</mi></mtd><mtd><mo>=</mo></mtd><mtd><mo>-</mo><mn>3</mn><mo> </mo><mo>(</mo><mi>Loại</mi><mo>)</mo></mtd></mtr></mtable></math></p>
<p>Vậy S = {6}.</p>
<p>d) Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mi>log</mi><mo>(</mo><mi>x</mi><mo>-</mo><mn>3</mn><mo>)</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇔</mo><mfenced open="{" close=""><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">x</mi><mo>-</mo><mn>3</mn><mo>></mo><mn>0</mn></mtd></mtr><mtr><mtd><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>7</mn><mo> </mo><mo>=</mo><mo> </mo><mi mathvariant="normal">x</mi><mo>-</mo><mn>3</mn></mtd></mtr></mtable><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">x</mi><mo>></mo><mn>3</mn></mtd></mtr><mtr><mtd><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>7</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>10</mn><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn></mtd></mtr></mtable></mfenced></mrow></mfenced><mo>⇔</mo><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>5</mn></math></p>
<p>Vậy S = {5}.</p>
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