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# 4.4.4E: Logarithmic Properties (Exercises)

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

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. $$x = 10$$