More Related Content Similar to 12 x1 t01 01 log laws (2013) (20) More from Nigel Simmons (20) 12 x1 t01 01 log laws (2013)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
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
25. e.g. (i) x log 5 125
5 x 125
x3
ii log x 343 3
26. e.g. (i) x log 5 125
5 x 125
x3
ii log x 343 3
x 3 343
27. e.g. (i) x log 5 125
5 x 125
x3
ii log x 343 3
x 3 343
x7
28. e.g. (i) x log 5 125
5 x 125
x3
iii Evaluate;
a) log 4 16
ii log x 343 3
x 3 343
x7
29. e.g. (i) x log 5 125
5 x 125
x3
iii Evaluate;
a) log 4 16
log 4 4 2
ii log x 343 3
x 3 343
x7
30. e.g. (i) x log 5 125
5 x 125
x3
iii Evaluate;
a) log 4 16
log 4 4 2
2 log 4 4
ii log x 343 3
x 3 343
x7
31. e.g. (i) x log 5 125
5 x 125
x3
iii Evaluate;
a) log 4 16
log 4 4 2
2 log 4 4
2
ii log x 343 3
x 3 343
x7
32. ii log x 343 3
e.g. (i) x log 5 125
5 x 125
x3
x 3 343
x7
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
x3
x 3 343
x7
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
x3
x 3 343
x7
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
x3
x 3 343
x7
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
x3
x 3 343
x7
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
x3
x 3 343
x7
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
x3
x 3 343
x7
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
x3
x 3 343
x7
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
x3
x 3 343
x7
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
x3
x 3 343
x7
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
x3
x 3 343
x7
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
x3
x 3 343
x7
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
x3
x 3 343
x7
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
x3
x 3 343
x7
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
x3
x 3 343
x7
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
x3
x 3 343
x7
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
x3
x 3 343
x7
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
x3
x 3 343
x7
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
x3
x 3 343
x7
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
x3
x 3 343
x7
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
54. iv 32 x 1
1
27
32 x 1 33
2 x 1 3
2 x 4
x 2
55. iv 32 x 1
1
27
32 x 1 33
2 x 1 3
2 x 4
x 2
v 2 x 9
56. iv 32 x 1
1
27
32 x 1 33
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 33
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 33
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 33
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 33
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