4.4E: Logarithmic Properties (Exercises)

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Jan 2, 2021
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- 4.4: Logarithmic Properties
- 4.5: Graphs of Logarithmic Functions

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section 4.4 exercise

Simplify to a single logarithm, using logarithm properties.

1. $\log _{3} \left(28\right)-\log _{3} \left(7\right)$

2. $\log _{3} \left(32\right)-\log _{3} \left(4\right)$

3. $-\log _{3} \left(\dfrac{1}{7} \right)$

4. $-\log _{4} \left(\dfrac{1}{5} \right)$

5. $\log _{3} \left(\dfrac{1}{10} \right)+\log _{3} \left(50\right)$

6. $\log _{4} \left(3\right)+\log _{4} (7)$

7. $\dfrac{1}{3} \log _{7} \left(8\right)$

8. $\dfrac{1}{2} \log _{5} \left(36\right)$

9. $\log \left(2x^{4} \right)+\log \left(3x^{5} \right)$

10. $\ln \left(4x^{2} \right)+\ln \left(3x^{3} \right)$

11. $\ln \left(6x^{9} \right)-\ln \left(3x^{2} \right)$

12. $\log \left(12x^{4} \right)-\log \left(4x\right)$

13. $2\log \left(x\right)+3\log \left(x+1\right)$

14. $3\log \left(x\right)+2\log \left(x^{2} \right)$

15. $\log \left(x\right)-\dfrac{1}{2} \log \left(y\right)+3\log \left(z\right)$

16. $2\log \left(x\right)+\dfrac{1}{3} \log \left(y\right)-\log \left(z\right)$

Use logarithm properties to expand each expression.

17. $\log \left(\dfrac{x^{15} y^{13} }{z^{19} } \right)$

18. $\log \left(\dfrac{a^{2} b^{3} }{c^{5} } \right)$

19. $\ln \left(\dfrac{a^{-2} }{b^{-4} c^{5} } \right)$

20. $\ln \left(\dfrac{a^{-2} b^{3} }{c^{-5} } \right)$

21. $\log \left(\sqrt{x^{3} y^{-4} } \right)$

22. $\log \left(\sqrt{x^{-3} y^{2} } \right)$

23. $\ln \left(y\sqrt{\dfrac{y}{1-y} } \right)$

24. $\ln \left(\dfrac{x}{\sqrt{1-x^{2} } } \right)$

25. $\log \left(x^{2} y^{3} \sqrt[{3}]{x^{2} y^{5} } \right)$

26. $\log \left(x^{3} y^{4} \sqrt[{7}]{x^{3} y^{9} } \right)$

Solve each equation for the variable.

27. $4^{4x-7} =3^{9x-6}$

28. $2^{2x-5} =7^{3x-7}$

29. $17\left(1.14\right)^{x} =19\left(1.16\right)^{x}$

30. $20\left(1.07\right)^{x} =8\left(1.13\right)^{x}$

31. $5e^{0.12t} =10e^{0.08t}$

32. $3e^{0.09t} =e^{0.14t}$

33. $\log _{2} \left(7x+6\right)=3$

34. $\log _{3} (2x+4)=2$

35. $2\ln \left(3{\rm x}\right)+3=1$

36. $4\ln \left(5x\right)+5=2$

37. $\log \left(x^{3} \right)=2$

38. $\log \left(x^{5} \right)=3$

39. $\log \left(x\right)+\log \left(x+3\right)=3$

40. $\log \left(x+4\right)+\log \left(x\right)=9$

41. $\log \left(x+4\right)-\log \left(x+3\right)=1$

42. $\log \left(x+5\right)-\log \left(x+2\right)=2$

43. $\log _{6} \left(x^{2} \right)-\log _{6} (x+1)=1$

44. $\log _{3} (x^{2} )-\log _{3} (x+2)=5$

45. $\log \left(x+12\right)=\log \left(x\right)+\log \left(12\right)$

46. $\log \left(x+15\right)=\log \left(x\right)+\log \left(15\right)$

47. $\ln \left(x\right)+\ln \left(x-3\right)=\ln \left(7x\right)$

48. $\ln \left(x\right)+\ln \left(x-6\right)=\ln \left(6x\right)$

Answer

1. $\text{log}_3 (4)$

3. $\text{log}_3 (7)$

5. $\text{log}_3 (5)$

7. $\text{log}_7 (2)$

9. $\text{log} (6x^9)$

11. $\text{ln} (2x^7)$

13. $\text{log}(x^2 (x + 1)^3)$

15. $\text{log} (\dfrac{xz^3}{\sqrt{y}})$

17. $15\text{log}(x) + 13 \text{log}(y) - 19 \text{log}(z)$

19. $-2\text{ln} (a) + 4\text{ln}(b) - 5 \text{ln}(c)$

21. $\dfrac{3}{2} \text{log}(x) - 2 \text{log}(y)$

23. $\text{ln}(y) + \dfrac{1}{2} (\text{ln} (y) - \text{ln} (1 - y))$

25. $\dfrac{8}{3} \text{log} (x) + \dfrac{14}{3} \text{log} (y)$

27. $x \approx -0.717$

29. $x \approx -6.395$

31. $t \approx 17.329$

33. $x = \dfrac{2}{7}$

35. $x \approx 0.123$

37. $x \approx 4.642$

39. $x \approx 30.158$

41. $x \approx -2.889$

43. $x \approx 6.873$ or $x \approx -0.873$

45. $x = \dfrac{12}{11} \approx 1.091$

47. $x = 10$

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