Bài 5: Phương trình mũ và phương trình lôgarit
Hướng dẫn giải Bài 4 (Trang 85 SGK Toán Giải Tích 12)
<p><strong>Bài 4 (Trang 85 SGK Toán Giải Tích 12):</strong></p>
<p>Giải các phương trình loogarit :</p>
<p>a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>log</mi><mfenced><mrow><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>+</mo><mi mathvariant="normal">x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mi>log</mi><mn>5</mn><mi mathvariant="normal">x</mi><mo> </mo><mo>+</mo><mo> </mo><mi>log</mi><mfrac><mn>1</mn><mrow><mn>5</mn><mi mathvariant="normal">x</mi></mrow></mfrac></math>;</p>
<p>b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>log</mi><mo>(</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo> </mo><mo>=</mo><mo> </mo><mi>log</mi><mn>8</mn><mi mathvariant="normal">x</mi><mo>-</mo><mi>log</mi><mn>4</mn><mi mathvariant="normal">x</mi></math>;</p>
<p>c) <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><msqrt><mn>2</mn></msqrt></msub><mi mathvariant="normal">x</mi><mo> </mo><mo>+</mo><mo> </mo><mn>4</mn><msub><mi>log</mi><mn>4</mn></msub><mi mathvariant="normal">x</mi><mo> </mo><mo>+</mo><mo> </mo><msub><mi>log</mi><mn>8</mn></msub><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>13</mn></math>.</p>
<p><strong><em>Hướng dẫn giải:</em></strong></p>
<p>a) Phương trình đã cho tương đương với hệ</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo> </mo><mo>+</mo><mo> </mo><mi mathvariant="normal">x</mi><mo> </mo><mo>-</mo><mo> </mo><mn>5</mn><mo> </mo><mo>></mo><mo> </mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>5</mn><mi mathvariant="normal">x</mi><mo> </mo><mo>></mo><mo> </mo><mn>0</mn><mo> </mo></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>log</mi><mfenced><mrow><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>+</mo><mi mathvariant="normal">x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn></mtd></mtr></mtable></mfenced><mo> </mo><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo> </mo><mo>+</mo><mo> </mo><mi mathvariant="normal">x</mi><mo> </mo><mo>-</mo><mo> </mo><mn>5</mn><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>0</mn></mtd></mtr><mtr><mtd><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo> </mo><mo>+</mo><mo> </mo><mi mathvariant="normal">x</mi><mo> </mo><mo>-</mo><mo> </mo><mn>5</mn><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo> </mo><mo>⇔</mo><mo> </mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">x</mi><mo> </mo><mo>></mo><mo> </mo><mfrac><mrow><msqrt><mn>21</mn></msqrt><mo> </mo><mo>-</mo><mo> </mo><mn>1</mn></mrow><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo> </mo><mo>+</mo><mo> </mo><mi mathvariant="normal">x</mi><mo> </mo><mo>-</mo><mo> </mo><mn>6</mn><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mspace linebreak="newline"/><mo>⇔</mo><mo> </mo><mfenced open="{" close=""><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">x</mi><mo> </mo><mo>></mo><mo> </mo><mfrac><mrow><msqrt><mn>21</mn></msqrt><mo> </mo><mo>-</mo><mo> </mo><mn>1</mn></mrow><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>3</mn><mo> </mo><mi>hoặc</mi><mo> </mo><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn></mtd></mtr></mtable><mo> </mo><mo>⇔</mo><mo> </mo><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn></mrow></mfenced></math></p>
<p>Vậy S = {2}.</p>
<p>b) Ta có <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>log</mi><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mi>log</mi><mn>8</mn><mi>x</mi><mo>-</mo><mi>log</mi><mn>4</mn><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇔</mo><mo> </mo><mfenced open="{" close=""><mrow><mtable columnalign="left"><mtr><mtd><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>-</mo><mo> </mo><mn>4</mn><mi>x</mi><mo> </mo><mo>-</mo><mo> </mo><mn>1</mn><mo> </mo><mo>></mo><mo> </mo><mn>0</mn></mtd></mtr><mtr><mtd><mi>x</mi><mo> </mo><mo>></mo><mo> </mo><mn>0</mn></mtd></mtr><mtr><mtd><mi>log</mi><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn><mi>log</mi><mfrac><mrow><mn>8</mn><mi>x</mi></mrow><mrow><mn>4</mn><mi>x</mi></mrow></mfrac></mtd></mtr></mtable><mo> </mo><mo>⇔</mo><mo> </mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>-</mo><mo> </mo><mn>4</mn><mi>x</mi><mo> </mo><mo>-</mo><mo> </mo><mn>1</mn><mo> </mo><mo>></mo><mo> </mo><mn>0</mn><mo> </mo></mtd></mtr><mtr><mtd><mi>x</mi><mo> </mo><mo>></mo><mo> </mo><mn>0</mn></mtd></mtr><mtr><mtd><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>-</mo><mo> </mo><mn>4</mn><mi>x</mi><mo> </mo><mo>-</mo><mo> </mo><mn>1</mn><mo> </mo><mo>=</mo><mo> </mo><mn>4</mn></mtd></mtr></mtable></mfenced></mrow></mfenced><mo> </mo><mspace linebreak="newline"/><mspace linebreak="newline"/><mo>⇔</mo><mo> </mo><mfenced open="{" close=""><mrow><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo> </mo><mo>></mo><mo> </mo><mn>2</mn><mo> </mo><mo>+</mo><mo> </mo><msqrt><mn>5</mn></msqrt></mtd></mtr><mtr><mtd><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>-</mo><mo> </mo><mn>4</mn><mi>x</mi><mo> </mo><mo>-</mo><mo> </mo><mn>5</mn><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn></mtd></mtr></mtable><mo> </mo><mo>⇔</mo><mo> </mo><mfenced open="{" close=""><mrow><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo> </mo><mo>></mo><mo> </mo><mn>2</mn><mo> </mo><mo>+</mo><mo> </mo><msqrt><mn>5</mn></msqrt></mtd></mtr><mtr><mtd><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>1</mn><mo> </mo><mi>h</mi><mi>o</mi><mi>ặ</mi><mi>c</mi><mo> </mo><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>5</mn></mtd></mtr></mtable><mo>⇔</mo><mo> </mo><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>5</mn></mrow></mfenced></mrow></mfenced></math></p>
<p>Vậy S = {5}</p>
<p>c) Với điều kiện x > 0, phương trình đã cho có thể viết lại thành </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><msup><mn>2</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></msub><mi mathvariant="normal">x</mi><mo>+</mo><mn>4</mn><mo> </mo><msub><mi>log</mi><msup><mn>2</mn><mn>2</mn></msup></msub><mi mathvariant="normal">x</mi><mo> </mo><mo>+</mo><mo> </mo><msub><mi>log</mi><msup><mn>2</mn><mn>3</mn></msup></msub><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>13</mn><mo>⇔</mo><mn>2</mn><msub><mi>log</mi><mn>2</mn></msub><mi mathvariant="normal">x</mi><mo>+</mo><mn>2</mn><mo> </mo><msub><mi>log</mi><mn>2</mn></msub><mi mathvariant="normal">x</mi><mo> </mo><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msub><mi>log</mi><mn>2</mn></msub><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>13</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇔</mo><msub><mi>log</mi><mn>2</mn></msub><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>3</mn><mo>⇔</mo><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><msup><mn>2</mn><mn>3</mn></msup><mo> </mo><mo>=</mo><mo> </mo><mn>8</mn></math></p>
<p>Vậy S = {8}</p>
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