Logarithms
Logarithms
Logarithms are the inverse of exponentials.
Logarithms
Logarithms are the inverse of exponentials.
If y  a x
Logarithms
Logarithms are the inverse of exponentials.
If y  a x then x  log a y
Logarithms
Logarithms are the inverse of exponentials.
If y  a x then x  log a y
If y  e x
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
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
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...
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...
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...
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...
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...
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 Laws
Log Laws
1 log a m  log a n  log a mn
Log Laws
1 log a m  log a n  log a mn
m
2 log a m  log a n  log a  
n
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
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...
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...
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...
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...
e.g. (i) x  log 5 125
e.g. (i) x  log 5 125
5 x  125
e.g. (i) x  log 5 125
5 x  125
x3
e.g. (i) x  log 5 125
5 x  125
x3

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

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

ii  log x 343  3
x 3  343
x7
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
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
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
...
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  3...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
iv  32 x 1 

1
27
iv  32 x 1 

1
27

32 x 1  33
iv  32 x 1 

1
27

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

1
27

32 x 1  33
2 x  1  3
2 x  4
x  2

v  2 x  9
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
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
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
l...
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
l...
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
l...
Upcoming SlideShare
Loading in...5
×

12 x1 t01 01 log laws (2013)

673

Published on

Published in: Education
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
673
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
26
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Transcript of "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
  1. A particular slide catching your eye?

    Clipping is a handy way to collect important slides you want to go back to later.

×