Logarithm equations
Logarithm Equations
Solve each of the following equations for x:
1. log(x) + log(x+9) = 1 2. log(x) – log(x + 3) = 1
3. log(x + 9) – log(x) = 1 4. log(2x + 1) – log(x – 9) = 1
5. log4(x + 3) + log4(x – 3) = 2 6. log8(x + 1) – log8(x) = log8(4)
7. log(x2 ) = (log x)2 8. (log3x)2 – log3(x2) = 3
9. log = 10. log(log x) = 2
11. log5 = 1 12. log + log = log(2–3 )
13. log6x + log6(x + 1) = 1 14. log9(x + 1) = + log9x
15. log2(x + 4) = 2 – log2(x + 1) 16. log(2x + 4) + log(x – 2) = 1
17. ln x + ln(x + 1) = ln 2 18. log(x + 3) – log(x – 2) = 2
19. log2(2x2 + 4) = 5 20. log(x – 6) + log(x + 3) = 1
21. log(x) – log(5) = log(2) – log(x – 3)
22. (ln x)3 = ln(x4) 23. log(6x + 5) – log(3) = log(2) – log(x)
24. (log x)3 = log(x4) 25. log(x2) – log(x – 1) = 1
26. log(x + 1) = log(2x2 + 3) – log(2x – 5) –1
27. (log x)2 – 3 log(x) + 2 = 0 28. log(27) + log(9 – 3) = log(x)
29. 3 + log2(3) + log2(x) = log2(96) 30. log(2x) – 2 log(x) = –1
31. – 5 + 4 = 0
32. [log(2x – 1)]2 – 3Log(2x – 1) – 10 = 0
33. 34. log2(3 – x) + log2(1 – x) = 3
35. Log3(2 – x) + Log3(4 – x) = 1 36. Log5(2 – x) + Log5(6 – x) = 1
37. Log2(3 – x) + Log2(7 – x) = 5 38. Log3(5 – x) + Log3(3 – x) = 1
39. Log2(3 – x) + Log2(6 – x) = 2 40. Log2(7 – x) + Log2(5 – x) = 3
41. Log6(7 – x) + Log6(1 – x) = 3 42. Log4(4 – x) + Log4(1 – x) = 1
43. log3x + log9x + log27x = 5.5 44. log(x – 3) + log(x + 6) = log(2) + log(5)
45. log(x – 4) + log(x + 3) = log(5x + 4)
46. ln(x2 +1) – ln(x2 – 2x + 1) = ln(5)
47. log5(x – 2) + 2 log5(x3 – 2) + log5(x – 2)–1 = 4
48. 2 log3(x – 2) + log3(x – 4)2 = 0
49. log2(x + 2)2 + log2(x + 10)2 = 4 log2(3)
50. log2 = log2 51. 2 log2 + log2 = 1
52. log3(5x – 2) – 2 log3 = 1 – log3(4)
53. log(3x – 2) – 2 = log(x + 2) – log(50)
54. log(10x2) log(x) = 1 55. 2 log9(x) + 9 logx(3) = 10
56. logx(125x) (log25x)2 = 1 57. Log2x + 2Logx8 = 5
58. Log3x + 2Logx9 = 5 59. Log3x – Logx27 = 2
60. Log4x + 4Logx64 = 8 61. Logx + 3Logx4 = 8
62. Logx + 4Logx27 = 14 63. log(log x) + log(log x3 – 2) = 0
64. log3x + 7(9 + 12x + 4x2) = 4 – log2x + 3(6x2 + 23x + 21)
65. x2 logx(27) log9(x) = x + 4 66. logx(2) – log4(x) + = 0
67. log.5xx2 – 14 log16xx3 + 40 log4x = 0
68. xlog(x) = 100x 69. xlog(x) =
70. (x + 1)log(x + 1) = 100(x + 1) 71. xlog(x) = 3x
72. 2 = 73. Solve for t: P = P0ekt
74. Solve for t: T = T0 + (T1 – T2)ekt 75. Solve for n: PVn = c
76. Solve for Q: LogaQ = Logay + b 77. Solve for x: Logax = b + Logab
78. Solve for y: Logay = Logab – a 79. Solve for p: LogbP = b – Logb(aP)
Answers
1. 1 2. no solution 3. 1 4. 11
5. 5 6. 7. 1; 100 8. 27;
9. 1; 104 10. 10100 11. 12.
13. 2 14. 15. 0 16. 3
17. 1 18. 2 19. 20. 7
21. 5 22. 1; e2; e–2 23. 24. 1; 100; .01
25. 5 ± 26. 27. 10; 100 28. 18
29. 4 30. 20 31. ;
32. .0505; (105 + 1) 33. 103 34. –1
35. 1 36. 1 37. 1 38. 2
39. 1 40. 3 41. –11 42. 0
43. 27 44. 4 45. 8 46. 2; 3
47. 3 48. 3 + , 3 49. –1 50. no solution
51. –17 52. 1 53. 2 54. ;
55. 3; 39 56. 5; 57. 8, 4 58. 81, 3
59. 3, 27 60. 64, 4096 61. 8, 2 62. 729, 3
63. 10; 64. – 65. 2 66. 8;
67. 1; 4; 68. 100; 69. 10; 100 70. 99; –
71. .1223 72.
12/06/06
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