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

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. 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. 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. 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. 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
7  log a x 
log b 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
5 x  125
x3
ii  log x 343  3

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

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

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. iv  32 x 1 
1
27

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

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

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. 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. 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. 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. 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. 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