4.4E: Logarithmic Properties (Exercises)

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. \(x = 10\)