algebralogarithm.pptx mathematics for intermediate

log

1. ARINDAM GOSWAMI DELHI PUBLIC SCHOOL GHAZIABAD LOG ANTI LOG

2. Introduction to Logarithm Fundamental Knowledge Its application 2

3. Definition of Logarithm 3

4. Ex a m ple 4

5. Properties of Logarithm 5

6. Things to Remember 6

7. Laws of Logarithm 7

8. Logarithm of a Product Rule Logarithm of the product of two numbers is equal to the sum of the logarithm of the numbers to the same base, i.e. log a (mn) = log a m + log a n 8

9. 9 Logarithm of a Product Rule Contd…

10. Logarithm of a Quotient Logarithm of a quotient of any two postive numbers to any real base (>1) is equal to the logarithm of the numerator – logarithm of the denominator to the same base i.e. log a (m/n) = log a m - log a n 10

11. 11 Logarithm of a Quotient Contd…

12. Logarithm of a power of a number The logarithm of a number to any rational index, to any real base (>1) is equal to the product of the index and the logarithm of the given number to the same base i.e. log a m n = nlog a m 12

13. 13 Logarithm of a power of a number Contd…

14. Change of Base 14

15. 15 Change of Base Contd…

16. Base Changing Result 16

17. Systems of Logarithm 17

18. Common Logarithms Natural Logarithms 18 Systems of Logarithms

19. Natural Logarithms The logarithm to the base e; where e is the sum of infinite series are called natural logarithms (e=2.7183 approx.). They are used in theoretical calculations 19

20. Common Logarithm Logarithm to the base 10 are called common logarithm. They are used in numerical (Practical) calculations. Thus when no base is mentioned in numerical calculations, the base is always understood to be 10. 20

21. Ex a m ple Power (+) of 10 (Positive Characteristic) Log a r ith m ic Form Power (-) of 10 (Negative C h a r ac t e r i s ti c) Log a r ith m ic Form 10 1 =10 log 10 10 =1 10 -1 = 0.1 log 10 0.1 =-1 10 2 =100 log 10 100=2 10 -2 = 0.01 log 10 0.01 =-2 10 3 =1000 log 10 1000=3 10 -3 = 0.001 log 10 0.001 =-3 21

22. Standard form of a number n Any positive decimal or number say ‘n’ can be written in the form of integral power of 10 say 10 p (where p is an integer) and a number m between 1 and 10. Therefore n = m x 10 p where p is an integer (positive, negative or zero) and m is such that 1≤m<10. This is called the standard form of n. Example- Write the Standard Form for the following (1) 259.8 (2) 25.98 (3) 0.2598 (4) 0.02598 22

23. Example – Continued 23

24. Characteristic and Mantissa The logarithm of a number consist of two parts, the whole part or integral part is called the characteristic and decimal part is called Mantissa. Mantissa is always positive and always less than 1. The characteristic is determined by bringing the given number n to the standard form n=m x 10 p , in which p (the power of 10) gives the characteristic and the mantissa is found from the logarithmic table. 24

25. Ex a m ple 25

26. Rules to find Characterstic 26

27. Rule 1 The characteristic of the logarithm of any number greater than 1 is positive and is one less than the number of digits to the left of the decimal point in the given number. Example: Consider the following table 27 Number Characteristic 48 1 3578 3 8.31

28. Rule 2 The characteristic of the logarithm of any number less than 1 is negative and numerically one more than the number of zeros to the right of the decimal point. If there is no zero then obviously it will be -1. Example: Consider the following table 28 Number Characteristic .6 -1 .09 -2 .00657 -3 .000852 -4

29. Mantissa The Mantissa of the common logarithm of a number can be found from a log-table. 29

30. What is Log Table 30

31. H o w t o u s e t h e L o g T a b l e t o f i nd Mantissa Remove the decimal point from the given number. Consider the first two digits. In horizontal row beginning with above two digits, read the number under column headed by 3 rd digit (from the left) of the number. To the number obtained above, add the number in the same horizontal line under the mean difference columns headed by 4 th digit (from the left) of the number. Then pre-fix the decimal point to the number obtained in 4 th point above. 31

32. Ex a m ple Suppose we have to find the log 125.6 Here characteristic is 3 – 1 = 2 For Mantissa, which is the positive decimal part. First remove decimal point, number becomes 1256 The first two digits are 12, the third is 5 and fourth is 6 32

33. Example- Continued Mantissa log 125.6 = 0.(0969+ 21) = 0.0990 = 2 + 0.0990 = 2.0990 33

34. Point to remember 34

35. Point to remember- Continued 35

36. Anti Logarithm The reverse process of finding the logarithm is called Antilogarithm i.e. to find the number. If x is the logarithm of a given number n with given base ‘a’ then n is called antilogarithm or antilog of x to that base. Mathematically, if log a n = x Then n = antilog x 36

37. Ex a m ple Find the number whose logarithm is 2.0239 From the Antilog Table For mantissa .023, the number = 1054 For mean difference 9, the number = 2 Therefore for mantissa .0239, the number = 1054 + 2 = 1056 37

38. Example- Continued Here the characteristic is 2 Therefore the number must have 3 digits in the integral part. Hence antilog 2.0239 = 105.6 38

39. Illustrations 39

40. Illustration 1 40

41. Illustration 2 41

42. Illustration 2 - Continued = 28 log 2 - 7 log 3 - 7 log 5 + 10 log 5 - 15 log 2 - 5 log 3 + 12 log 3 - 12 log 2 - 3 log 5 = (28 - 15 - 12) log 2 + (- 7 - 5 + 12) log 3 + (- 7 + 10 - 3) log 5 = log 2. = R.H.S 42

43. Illustration 3 The value of log 2 [log 2 {log 3 (log 3 27 3 )}] is (a) 1 (b) 2 (c) 0 (d) None of these Solution : Given expression = log 2 [log 2 {log 3 (3log 3 27 )}] = log 2 [log 2 {log 3 (31og 3 3 3 )} ] = log 2 [log 2 {log 3 (9log 3 3)}] 43

44. Illustration 3 – Continued (as log 3 3 = 1) = log 2 [log 2 {log 3 (9X1)}] = log 2 [log 2 {log 3 3 2 }] = log 2 [log 2 (2log 3 3)] = log 2 [log 2 2] = log 2 1 = 44

45. Illustration 4 45

46. Illustration 4 – Continued 46

47. Illustration – 5 47

48. Illustration – 5 - Continued L.H.S. = K (y – z) (y 2 + z 2 + yz) + K (z – x) (z 2 + x 2 +xz) + K (x – y) (x 2 + y 2 + xy) = K (y 3 – z 3 ) + K (z 3 – x 3 ) + K (x 3 – z 3 ) = K (y 3 – z 3 + z 3 – x 3 + x 3 – y 3 ) = K. = = R.H.S. 48

49. Illustration – 6 49

50. Illustration – 6 – Continued 50

51. Illustration – 7 51

52. Illustration – 7 – Continued 52

53. Illustration – 8 53

54. Illustration – 9 54

55. Illustration – 10 55

56. Illustration – 10 – Continued 56

57. Illustration – 11 57

58. Illustration – 11 – Continued loga + logb + logc = ky – kz+ kz – kx + kx – ky log(abc) = 0 log(abc) = log1 abc = 1 58

59. Illustration – 12 59

60. Illustration – 12 – Continued 60

61. Illustration – 13 61

62. Illustration – 13 – Continued 62

63. Illustration – 14 63

64. Illustration – 14 – Continued 64

65. Illustration – 15 65

66. Illustration – 15 - Continued 66

67. Illustration – 16 67

68. Illustration – 16 - Continued 68

69. Illustration – 17 69

70. Illustration – 17 – Continued 70

71. Illustration 18 log b (a) . log c (b) . log a (c) is equal to (c) -1 (d) None of these (a) 0 (b) 1 Solution: log b (a) . log c (b) . log a (c) = log c a . log a c = log a a =1 71

72. Illustration 19 a logb – logc . b logc – loga . c loga – logb has a value of (a) 1 (b) 0 (c) -1 (d) None of these Solution: Let x = a logb – logc . b logc – loga . c loga – logb Taking log on both sides, we get logx = log(a logb – logc . b logc – loga . c loga – logb ) = loga logb – logc + logb logc – loga + logc loga – logb 72 ∴

73. Illustration 19 – Continued 73

74. Illustration 20 74

75. Illustration 20 - Continued 75

76. Making graph

77. Thank You! 76