Logaritmes

Recorda aplicar la definició i les propietats de logaritmes.

Exercici 1. Calcula el logaritme.

  1. \(\log_2 8\) =
  2. \(\log_3 9\) =
  3. \(\log_4 2\) =
  4. \(\log_{27} 3\) =
  5. \(\log_5 0.2\) =
  6. \(\log_2 0.25\) =
  7. \(\log_{0.5} 16\) =
  8. \(\log_{0.1} 100\) =
  9. \(\log_a \sqrt[3]{a^2}\) =
  10. \(\log_{\sqrt{2}} 2\) =
  11. \(\log_4 64\) =
  12. \(\log_{2\sqrt{2}} 0.25\) =
  13. \(\log_{\sqrt{2}} 32\) =
  14. \(\log_{1/3} \sqrt[3]{9}\) =
  15. \(\ln \sqrt[5]{e^2}\) =
  16. \(\log 0.0001\) =
  17. \(\log 0\) =
  18. \(\log_5 5\sqrt{5}\) =
  19. \(\log_3 \frac{\sqrt[4]{3}}{\sqrt{27}}\) =
  20. \(\log_4 \frac{1}{\sqrt[3]{1024}}\) =
  21. \(\log_3 27 + \log_3 1\) =
  22. \(\log_5 25 - \log_5 5\) =
  23. \(\log_4 64 + \log_8 64\) =
  24. \(\log 0.1 - \log 0.01\) =
  25. \(\log 5 + \log 20\) =
  26. \(\log 2 - \log 0.2\) =
  27. \(\log 32 / \log 2\) =
  28. \(\log 3 / \log 81\) =
  29. \(\log_2 3 \times \log_3 4\) =
  30. \(\log_9 25 \div \log_3 5\) =
Solucions
  1. 3
  2. 2
  3. 1/2
  4. 1/3
  5. -1
  6. -2
  7. -4
  8. -2
  9. 2/3
  10. 2
  11. 3
  12. -4/3
  13. 10
  14. -2/3
  15. 2/5
  16. -4
  17. 7
  18. 3/2
  19. -5/4
  20. -5/3
  21. 3
  22. 1
  23. 5
  24. 1
  25. 2
  26. 1
  27. 5
  28. 1/4
  29. 2
  30. 1

Exercici 2. Determina el valor de x.

  1. \(\log_3 81 = x\)
  2. \(\log_5 0.2 = x\)
  3. \(\log_2 16 = x^3/2\)
  4. \(\log_2 x = -3\)
  5. \(\log_7 x = 3\)
  6. \(\log_x 125 = 3\)
  7. \(\log_x 25 = -2\)
  8. \(\log_{2x+3} 81 = 2\)
  9. \(x + 2 = 10^{\log 5}\)
  10. \(x = 10^{4\log 2}\)
  11. \(x = \frac{\log 8}{\log 2}\)
  12. \(\log_\frac{9}{16} x = \frac{3}{2}\)
  13. \(\log_4 64 = \frac{2x - 1}{3}\)
  14. \(\log_6 [4(x - 1)] = 2\)
  15. \(\log_8 [2(x^3 + 5)] = 2\)
  16. \(x = \frac{\log 625}{\log 125}\)
  17. \(\frac{\log (x + 1)}{\log (x - 1)} = 2\)
  18. \(\frac{\log (x - 7)}{\log (x - 1)} = 0.5\)
  19. \(\log_7 7x = 2\)
  20. \(\log_x \frac{1}{3} = -\frac{1}{2}\)
  21. \(\log_x e = -3\)
  22. \(\log_x 0.015625 = -3\)
  23. \(\log_7 x^4 = 2\)
  24. \(\log_{\frac{1}{8}} x = \frac{1}{3}\)
Solucions
  1. 4
  2. -1
  3. 2
  4. 1/8
  5. 343
  6. 5
  7. 1/5
  8. 3
  9. 3
  10. 16
  11. 3
  12. 27/64
  13. 5
  14. 10
  15. 3
  16. 4/3
  17. 3
  18. 10
  19. 7
  20. 9
  21. \(e^{-1/3}\)
  22. 4
  23. \(\pm \sqrt{7}\)
  24. 1/2

Exercici 3. Calcula el valor de les expressions següents.

  1. \(\log_2 \frac{\sqrt[6]{64} \cdot 4^2}{2^5 \cdot \sqrt[3]{512}} =\)
  2. \(\log_3 \frac{27 \cdot \sqrt{729}}{81 \cdot \sqrt[3]{27}} =\)
  3. \(\log_5 \frac{25 \cdot \sqrt[4]{625}}{125} =\)
  4. \(\log_7 \frac{49 \cdot \sqrt[3]{343}}{\sqrt{2401}} =\)
  5. \(\log \left( \frac{0.01 \cdot \sqrt[3]{100}}{10^{-1} \cdot 0.1} \right) =\)
Solucions
  1. -3
  2. 1
  3. 0
  4. 1
  5. 2/3

Exercici 4. Redueix a un sol logaritme.

  1. \(\log a + \log b\) =
  2. \(\log x - \log y\) =
  3. \(\tfrac12\log x + \tfrac12\log y\) =
  4. \(\log a - \log x - \log y\) =
  5. \(\log p + \log q - \log r - \log s\) =
  6. \(\log 2 + \log 3 + \log 4\) =
  7. \(\tfrac13\log a - \tfrac12\log b - \tfrac12\log c\) =
  8. \(\tfrac32\log a + \tfrac52\log b\) =
  9. \(\log a + \tfrac12\log b - 2\log c\) =
  10. \(\log(a+b) + \log(a-b)\) =
  11. \(\tfrac12\log x - \tfrac13\log y + \tfrac14\log z\) =
  12. \(\log(a-b) - \log 3\) =
  13. \(\log a \;-\; 4\log b \;+\; \tfrac15\bigl(\log c - 2\log d\bigr)\) =
  14. \(\tfrac{p}{n}\log a + \tfrac{q}{n}\log b\) =
  15. \(\log_{a}(a\,c) + \log_{d}(d^{3}) + \log_{b} b - \log_{a} c\) =
Solucions
  1. \(\log(a\cdot b)\)
  2. \(\log\left(\frac{x}{y}\right)\)
  3. \(\log\sqrt{xy}\)
  4. \(\log\left(\frac{a}{xy}\right)\)
  5. \(\log\left(\frac{\rho\cdot q}{r\cdot s}\right)\)
  6. \(\log 24\)
  7. \(\log\left(\frac{\sqrt[3]{a}}{\sqrt{b\cdot c}}\right)\)
  8. \(\log\sqrt{a^{3}b^{5}}\)
  9. \(\log\left(\frac{a\sqrt{b}}{c^{2}}\right)\)
  10. \(\log(a^{2}-b^{2})\)
  11. \(\log\left(\frac{\sqrt{x}}{\sqrt[3]{y}\,\sqrt[4]{z}}\right)\)
  12. \(\log\left(\frac{a-b}{3}\right)\)
  13. \(\log\left(\frac{a}{b^{4}}\sqrt{\frac{c}{d^{2}}}\right)\)
  14. \(\log\sqrt[n]{a^{\rho}\cdot b^{\sigma}}\)
  15. \(5\)

Exercici 5. Sabent que log 2= 0,3 i que log 3= 0,48, calcula els logaritmes següents.

  1. \(\log 4\) =
  2. \(\log 5\) =
  3. \(\log 6\) =
  4. \(\log 8\) =
  5. \(\log 12\) =
  6. \(\log 15\) =
  7. \(\log 18\) =
  8. \(\log 24\) =
  9. \(\log 25\) =
  10. \(\log 30\) =
  11. \(\log 36\) =
  12. \(\log 40\) =
  13. \(\log 45\) =
  14. \(\log 60\) =
  15. \(\log 72\) =
  16. \(\log 75\) =
Solucions
  1. 0,6
  2. 0,7
  3. 0,78
  4. 0,9
  5. 1,08
  6. 1,18
  7. 1,26
  8. 1,38
  9. 1,4
  10. 1,48
  11. 1,56
  12. 1,6
  13. 1,66
  14. 1,78
  15. 1,86
  16. 1,88

Exercici 6. Expressa en funció de log 2 i de log 3 les expressions següents.

  1. \(\log 14.4\) =
  2. \(\log 0.048\) =
  3. \(\log 3600\) =
  4. \(\log \sqrt{5.76}\) =
  5. \(\log \frac{\sqrt{5.4}}{12.8}\) =
  6. \(\log \frac{1}{6561}\) =
  7. \(\log \left(\sqrt{3^2 \cdot \sqrt{16}}\right)\) =
  8. \(\log \sqrt[3]{\frac{9}{2}}\) =
Solucions
  1. \(4\log 2 + 2\log 3 - 1\)
  2. \(4\log 2 + \log 3 - 3\)
  3. \(2(1 + \log 2 + \log 3)\)
  4. \(3\log 2 + \log 3 - 1\)
  5. \(\tfrac12 \left(1 + 3\log 3 - 13\log 2\right)\)
  6. \(-8\log 3\)
  7. \(\tfrac92 \log 2 - 1\)
  8. \(\tfrac13(2\log 3 - \log 2)\)