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# 12 x1 t01 01 log laws (2013)

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### 12 x1 t01 01 log laws (2013)

1. 1. Logarithms
2. 2. Logarithms Logarithms are the inverse of exponentials.
3. 3. Logarithms Logarithms are the inverse of exponentials. If y  a x
4. 4. Logarithms Logarithms are the inverse of exponentials. If y  a x then x  log a y
5. 5. Logarithms Logarithms are the inverse of exponentials. If y  a x then x  log a y If y  e x
6. 6. Logarithms Logarithms are the inverse of exponentials. If y  a x then x  log a y If y  e x then x  log e y
7. 7. Logarithms Logarithms are the inverse of exponentials. If y  a x then x  log a y If y  e x then x  log e y x  ln y
8. 8. Logarithms Logarithms are the inverse of exponentials. If y  a x then x  log a y If y  e x then x  log e y x  ln y x  log y
9. 9. Logarithms Logarithms are the inverse of exponentials. If y  a x then x  log a y If y  e x then x  log e y x  ln y x  log y log base e is known as the natural logarithm.
10. 10. Logarithms Logarithms are the inverse of exponentials. If y  a x then x  log a y If y  e x then x  log e y x  ln y x  log y log base e is known as the natural logarithm. y  log a x y 1 x a  1
11. 11. Logarithms Logarithms are the inverse of exponentials. If y  a x then x  log a y If y  e x then x  log e y x  ln y x  log y log base e is known as the natural logarithm. y  log a x y 1 a  1 x y  log a x 0  a  1
12. 12. Logarithms Logarithms are the inverse of exponentials. If y  a x then x  log a y If y  e x then x  log e y x  ln y x  log y y  log a x y domain : x  0 log base e is known as the natural logarithm. 1 a  1 x y  log a x 0  a  1
13. 13. Logarithms Logarithms are the inverse of exponentials. If y  a x then x  log a y If y  e x then x  log e y x  ln y x  log y y  log a x y domain : x  0 range : all real y log base e is known as the natural logarithm. 1 a  1 x y  log a x 0  a  1
14. 14. Log Laws
15. 15. Log Laws 1 log a m  log a n  log a mn
16. 16. Log Laws 1 log a m  log a n  log a mn m 2 log a m  log a n  log a   n
17. 17. Log Laws 1 log a m  log a n  log a mn m 2 log a m  log a n  log a   n 3 log a m n  n log a m
18. 18. Log Laws 1 log a m  log a n  log a mn m 2 log a m  log a n  log a   n 3 log a m n  n log a m 4 log a 1  0
19. 19. Log Laws 1 log a m  log a n  log a mn m 2 log a m  log a n  log a   n 3 log a m n  n log a m 4 log a 1  0 5 log a a  1
20. 20. Log Laws 1 log a m  log a n  log a mn m 2 log a m  log a n  log a   n 3 log a m n  n log a m 4 log a 1  0 5 log a a  1 6 a log x  x a
21. 21. Log Laws 1 log a m  log a n  log a mn m 2 log a m  log a n  log a   n 3 log a m n  n log a m 4 log a 1  0 5 log a a  1 6 a log x  x a 7  log a x  log b x log b a
22. 22. e.g. (i) x  log 5 125
23. 23. e.g. (i) x  log 5 125 5 x  125
24. 24. e.g. (i) x  log 5 125 5 x  125 x3
25. 25. e.g. (i) x  log 5 125 5 x  125 x3 ii  log x 343  3
26. 26. e.g. (i) x  log 5 125 5 x  125 x3 ii  log x 343  3 x 3  343
27. 27. e.g. (i) x  log 5 125 5 x  125 x3 ii  log x 343  3 x 3  343 x7
28. 28. e.g. (i) x  log 5 125 5 x  125 x3 iii  Evaluate; a) log 4 16 ii  log x 343  3 x 3  343 x7
29. 29. e.g. (i) x  log 5 125 5 x  125 x3 iii  Evaluate; a) log 4 16  log 4 4 2 ii  log x 343  3 x 3  343 x7
30. 30. e.g. (i) x  log 5 125 5 x  125 x3 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 ii  log x 343  3 x 3  343 x7
31. 31. e.g. (i) x  log 5 125 5 x  125 x3 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 ii  log x 343  3 x 3  343 x7
32. 32. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 b) 6 2log 6 3
33. 33. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 b) 6 2log 6 3 6 log 6 32
34. 34. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 b) 6 2log 6 3 log 6 32 6  32
35. 35. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 b) 6 2log 6 3 log 6 32 6  32 9
36. 36. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 b) 6 2log 6 3 log 6 32 6  32 9 c) log 216  log 2 8
37. 37. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 b) 6 2log 6 3 log 6 32 6  32 9 c) log 216  log 2 8  log 2 128
38. 38. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 b) 6 2log 6 3 log 6 32 6  32 9 c) log 216  log 2 8  log 2 128  log 2 27
39. 39. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 b) 6 2log 6 3 log 6 32 6  32 9 c) log 216  log 2 8  log 2 128  log 2 27 7
40. 40. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 b) 6 2log 6 3 log 6 32 6  32 9 d) log10125  log10 32  log10 4 c) log 216  log 2 8  log 2 128  log 2 27 7
41. 41. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 b) 6 2log 6 3 log 6 32 6  32 9 d) log10125  log10 32  log10 4 125  32   log10    4  c) log 216  log 2 8  log 2 128  log 2 27 7
42. 42. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 b) 6 2log 6 3 log 6 32 6  32 9 d) log10125  log10 32  log10 4 125  32   log10    4   log10 1000 c) log 216  log 2 8  log 2 128  log 2 27 7
43. 43. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 b) 6 2log 6 3 log 6 32 6  32 9 d) log10125  log10 32  log10 4 125  32   log10    4   log10 1000 3 c) log 216  log 2 8  log 2 128  log 2 27 7
44. 44. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 c) log 216  log 2 8 b) 6 2log 6 3  log 2 128  log 2 27 7 log 6 32 6  32 9 d) log10125  log10 32  log10 4 125  32   log10    4   log10 1000 3 e) log 7 8 log 7 2
45. 45. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 c) log 216  log 2 8 b) 6 2log 6 3  log 2 128  log 2 27 7 log 6 32 6  32 9 d) log10125  log10 32  log10 4 125  32   log10    4   log10 1000 3 e) log 7 8 log 7 2  log 2 8
46. 46. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 c) log 216  log 2 8 b) 6 2log 6 3  log 2 128  log 2 27 7 log 6 32 6  32 9 d) log10125  log10 32  log10 4 125  32   log10    4   log10 1000 3 e) log 7 8 log 7 2  log 2 8 3
47. 47. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 c) log 216  log 2 8 b) 6 2log 6 3  log 2 128  log 2 27 7 log 6 32 6  32 9 d) log10125  log10 32  log10 4 125  32   log10    4   log10 1000 3 e) log 7 8 log 7 2  log 2 8 3 f) log 2 1 8
48. 48. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 c) log 216  log 2 8 b) 6 2log 6 3  log 2 128  log 2 27 7 log 6 32 6  32 9 d) log10125  log10 32  log10 4 125  32   log10    4   log10 1000 3 e) log 7 8 log 7 2  log 2 8 3 f) log 2 1 8 1 1  log 2 2 8
49. 49. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 c) log 216  log 2 8 b) 6 2log 6 3  log 2 128  log 2 27 7 log 6 32 6  32 9 d) log10125  log10 32  log10 4 125  32   log10    4   log10 1000 3 e) log 7 8 log 7 2  log 2 8 3 f) log 2 1 8 1 1  log 2 2 8
50. 50. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 c) log 216  log 2 8 b) 6 2log 6 3  log 2 128  log 2 27 7 log 6 32 6  32 9 d) log10125  log10 32  log10 4 125  32   log10    4   log10 1000 3 e) log 7 8 log 7 2  log 2 8 3 f) log 2 1 8 1 1  log 2 2 8 1   3 2
51. 51. ii  log x 343  3 e.g. (i) x  log 5 125 5 x  125 x3 x 3  343 x7 iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2 c) log 216  log 2 8 b) 6 2log 6 3  log 2 128  log 2 27 7 log 6 32 6  32 9 d) log10125  log10 32  log10 4 125  32   log10    4   log10 1000 3 e) log 7 8 log 7 2  log 2 8 3 f) log 2 1 8 1 1  log 2 2 8 1   3 2 3  2
52. 52. iv  32 x 1  1 27
53. 53. iv  32 x 1  1 27 32 x 1  33
54. 54. iv  32 x 1  1 27 32 x 1  33 2 x  1  3 2 x  4 x  2
55. 55. iv  32 x 1  1 27 32 x 1  33 2 x  1  3 2 x  4 x  2 v  2 x  9
56. 56. iv  32 x 1  1 27 32 x 1  33 2 x  1  3 2 x  4 x  2 v  2 x  9 log 2 x  log 9
57. 57. iv  32 x 1  1 27 32 x 1  33 2 x  1  3 2 x  4 x  2 v  2 x  9 log 2 x  log 9 x log 2  log 9
58. 58. iv  32 x 1  1 27 32 x 1  33 2 x  1  3 2 x  4 x  2 v  2 x  9 log 2 x  log 9 x log 2  log 9 log 9 x log 2
59. 59. iv  32 x 1  1 27 32 x 1  33 2 x  1  3 2 x  4 x  2 v  2 x  9 log 2 x  log 9 x log 2  log 9 log 9 x log 2 x  3.17 (to 2 dp)
60. 60. iv  32 x 1  1 27 32 x 1  33 2 x  1  3 2 x  4 x  2 v  2 x  9 log 2 x  log 9 x log 2  log 9 log 9 x log 2 x  3.17 (to 2 dp) Exercise 12A; 2, 3aceg, 4bdfh, 5ab, 6ab, 7ac, 8bdh, 9ac, 14, 18* Exercise 6B; 8