Ratio and Proportion, Indices and Logarithm Chapter 1 Paper 4: Quantitative Aptitude-Statistics. Ms. Ritu Gupta B.A. (Hons.) Maths and MA (Maths)

Transcription

1 Ratio and Proportion, Indices and Logarithm Chapter 1 Paper 4: Quantitative Aptitude-Statistics Ms. Ritu Gupta B.A. (Hons.) Maths and MA (Maths)

2 2 Introduction to Logarithm Fundamental Knowledge Its application

3 Definition of Logarithm 3

4 Example 4

5 Properties of Logarithm 5

6 Things to Remember 6

7 Laws of Logarithm 7

8 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

9 Logarithm of a Product Rule Contd 9

10 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

11 Logarithm of a Quotient Contd 11

12 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

13 Logarithm of a power of a number Contd 13

14 Change of Base 14

15 Change of Base Contd 15

16 Base Changing Result 16

17 Systems of Logarithm 17

18 18 Systems of Logarithms Natural Logarithms Common Logarithms

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

20 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.

21 21 Example Power (+) of 10 (Positive Characteristic) Logarithmic Form Power (-) of 10 (Negative Characteristic) Logarithmic Form 10 1 =10 log = = 0.1 log = =100 log = = 0.01 log = =1000 log = = log =-3

22 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) (2) (3) (4)

23 Example Continued 23

24 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.

25 Example 25

26 Rules to find Characterstic 26

27 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 Number Characteristic

28 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 Number Characteristic

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

30 What is Log Table 30

31 31 How to use the Log Table to find Mantissa 1. Remove the decimal point from the given number. 2. Consider the first two digits. 3. In horizontal row beginning with above two digits, read the number under column headed by 3 rd digit (from the left) of the number. 4. 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. 5. Then pre-fix the decimal point to the number obtained in 4 th point above.

32 32 Example Suppose we have to find the log 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

33 33 Example- Continued Mantissa = 0.( ) = log = =

34 Point to remember 34

35 Point to remember- Continued 35

36 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

37 37 Example Find the number whose logarithm is From the Antilog Table For mantissa.023, the number = 1054 For mean difference 9, the number = 2 Therefore for mantissa.0239, the number = = 1056

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

39 Illustrations 39

40 Illustration 1 40

41 Illustration 2 41

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

43 43 Illustration 3 The value of log 2 [log 2 {log 3 (log )}] 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 )} ] = log 2 [log 2 {log 3 (9log 3 3)}]

44 44 Illustration 3 Continued = log 2 [log 2 {log 3 (9X1)}] (as log 3 3 = 1) = log 2 [log 2 {log }] = log 2 [log 2 (2log 3 3)] = log 2 [log 2 2] = log 2 1 = 0

45 Illustration 4 45

46 Illustration 4 Continued 46

47 Illustration 5 47

48 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. 0 = 0 = R.H.S.

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

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

72 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. cloga 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 + logcloga logb

73 Illustration 19 Continued 73

74 Illustration 20 74

75 Illustration 20 - Continued 75

76 Thank You! 76