1. Logarithms

2. Logarithms
Logarithms are the inverse of exponentials.

3. Logarithms
Logarithms are the inverse of exponentials.
If y  a x

4. Logarithms
Logarithms are the inverse of exponentials.
If y  a x then x  log a y

5. Logarithms
Logarithms are the inverse of exponentials.
If y  a x then x  log a y
If y  e x

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. 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. 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. 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
log base e is known as
x  ln y
the natural logarithm.
x  log y

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
log base e is known as
x  ln y
the natural logarithm.
x  log y
y y  log a x a  1
1 x

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
log base e is known as
x  ln y
the natural logarithm.
x  log y
y y  log a x a  1
1 x
y  log a x 0  a  1

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
log base e is known as
x  ln y
the natural logarithm.
x  log y
y y  log a x a  1
1 x
domain : x  0
y  log a x 0  a  1

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
log base e is known as
x  ln y
the natural logarithm.
x  log y
y y  log a x a  1
1 x
domain : x  0
range : all real y y  log a x 0  a  1

14. Log Laws

15. Log Laws
1 log a m  log a n  log a mn

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. 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. 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. 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. 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. 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
log b x
7  log a x 
log b a

22. e.g. (i) x  log 5 125

23. e.g. (i) x  log 5 125
5 x  125

24. e.g. (i) x  log 5 125
5 x  125
x3

25. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125
x3

26. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3

27. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7

28. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16

29. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16
 log 4 4 2

30. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16
 log 4 4 2
 2 log 4 4

31. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16
 log 4 4 2
 2 log 4 4
2

32. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3
 log 4 4 2
 2 log 4 4
2

33. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3
 log 4 4 2 6 log 6 32
 2 log 4 4
2

34. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3
 log 4 4 2 6 log 6 32
 2 log 4 4  32
2

35. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3
 log 4 4 2 6 log 6 32
 2 log 4 4  32
2 9

36. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8
 log 4 4 2 6 log 6 32
 2 log 4 4  32
2 9

37. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8
 log 4 4 2 6 log 6 32
 log 2 128
 2 log 4 4  32
2 9

38. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8
 log 4 4 2 6 log 6 32
 log 2 128
 2 log 4 4  32  log 2 27
2 9

39. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8
 log 4 4 2 6 log 6 32
 log 2 128
 2 log 4 4  32  log 2 27
2 9 7

40. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8
 log 4 4 2 6 log 6 32
 log 2 128
 2 log 4 4  32  log 2 27
2 9 7
d) log10125  log10 32  log10 4

41. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8
 log 4 4 2 6 log 6 32
 log 2 128
 2 log 4 4  32  log 2 27
2 9 7
d) log10125  log10 32  log10 4
125  32 
 log10 
 
 4 

42. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8
 log 4 4 2 6 log 6 32
 log 2 128
 2 log 4 4  32  log 2 27
2 9 7
d) log10125  log10 32  log10 4
125  32 
 log10 
 
 4 
 log10 1000

43. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8
 log 4 4 2 6 log 6 32
 log 2 128
 2 log 4 4  32  log 2 27
2 9 7
d) log10125  log10 32  log10 4
125  32 
 log10 
 
 4 
 log10 1000
3

44. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8
 log 4 4 2 6 log 6 32
 log 2 128
 2 log 4 4  32  log 2 27
2 9 7
d) log10125  log10 32  log10 4 log 7 8
e)
125  32  log 7 2
 log10 
 
 4 
 log10 1000
3

45. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8
 log 4 4 2 6 log 6 32
 log 2 128
 2 log 4 4  32  log 2 27
2 9 7
d) log10125  log10 32  log10 4 log 7 8
e)
125  32  log 7 2
 log10 
 
 4   log 2 8
 log10 1000
3

46. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8
 log 4 4 2 6 log 6 32
 log 2 128
 2 log 4 4  32  log 2 27
2 9 7
d) log10125  log10 32  log10 4 log 7 8
e)
125  32  log 7 2
 log10 
 
 4   log 2 8
 log10 1000 3
3

47. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8
 log 4 4 2 6 log 6 32
 log 2 128
 2 log 4 4  32  log 2 27
2 9 7 1
f) log 2
log 7 8 8
d) log10125  log10 32  log10 4 e)
125  32  log 7 2
 log10 
 
 4   log 2 8
 log10 1000 3
3

48. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8
 log 4 4 2 6 log 6 32
 log 2 128
 2 log 4 4  32  log 2 27
2 9 7 1
f) log 2
log 7 8 8
d) log10125  log10 32  log10 4 e) 1 1
log 7 2  log 2
125  32 
 log10 
 
2 8
 4   log 2 8
 log10 1000 3
3

49. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8
 log 4 4 2 6 log 6 32
 log 2 128
 2 log 4 4  32  log 2 27
2 9 7 1
f) log 2
log 7 8 8
d) log10125  log10 32  log10 4 e) 1 1
log 7 2  log 2
125  32 
 log10 
 
2 8
 4   log 2 8
 log10 1000 3
3

50. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8
 log 4 4 2 6 log 6 32
 log 2 128
 2 log 4 4  32  log 2 27
2 9 7 1
f) log 2
log 7 8 8
d) log10125  log10 32  log10 4 e) 1 1
log 7 2  log 2
125  32 
 log10 
 
2 8
 4   log 2 8 1
 log10 1000   3
3 2
3

51. e.g. (i) x  log 5 125 ii  log x 343  3
5 x  125 x 3  343
x3 x7
iii  Evaluate;
a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8
 log 4 4 2 6 log 6 32
 log 2 128
 2 log 4 4  32  log 2 27
2 9 7 1
f) log 2
log 7 8 8
d) log10125  log10 32  log10 4 e) 1 1
log 7 2  log 2
125  32 
 log10 
 
2 8
 4   log 2 8 1
 log10 1000   3
3 2
3 3

2

52. 1
iv  32 x 1 
27

53. 1
iv  32 x 1 
27
32 x 1  33

54. 1
iv  32 x 1 
27
32 x 1  33
2 x  1  3
2 x  4
x  2

55. 1
iv  32 x 1  v  2 x  9
27
32 x 1  33
2 x  1  3
2 x  4
x  2

56. 1
iv  32 x 1  v  2 x  9
27
log 2 x  log 9
32 x 1  33
2 x  1  3
2 x  4
x  2

57. 1
iv  32 x 1  v  2 x  9
27
log 2 x  log 9
32 x 1  33
x log 2  log 9
2 x  1  3
2 x  4
x  2

58. 1
iv  32 x 1  v  2 x  9
27
log 2 x  log 9
32 x 1  33
x log 2  log 9
2 x  1  3
log 9
2 x  4 x
log 2
x  2

59. 1
iv  32 x 1  v  2 x  9
27
log 2 x  log 9
32 x 1  33
x log 2  log 9
2 x  1  3
log 9
2 x  4 x
log 2
x  2 x  3.17 (to 2 dp)

60. 1
iv  32 x 1  v  2 x  9
27
log 2 x  log 9
32 x 1  33
x log 2  log 9
2 x  1  3
log 9
2 x  4 x
log 2
x  2 x  3.17 (to 2 dp)
Exercise 12A; 2, 3aceg, 4bdfh, 5ab, 6ab, 7ac, 8bdh, 9ac, 14, 18*
Exercise 6B; 8