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INDICES AND LOGARITHMS

ADDITIONAL

MATHEMATICS

MODULE 8

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CHAPTER 5 : INDICES AND LOGARITHMS

CONTENTS PAGE

5.0 INDICES AND LOGARITHM

5.1 Concept Map 2

5.2 Indices and laws of Indices 3

Exercise 5.2

5.3 Simplify Algebraic Expressions 4 - 5

Exercise 5.3

5.4 Equation Involving Indices 5 - 7

Exercise 5.4

5.5 Past Year Questions 8

5.6 Assessment 9 - 10

Answers 11


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5.1 CONCEPTUAL MAP

Indices

an index

base

Logarithms

If N = ax, then loga N = x

Fractional indices

a) a1/n

= ___________

b) am/n = ___________

= ___________

Laws of indices

a) a

m

x a

n

= _________

b) am an = _________

c) (am)n = _________

Integer indices

a) an = a x a x a x a x a

n factors

b) Negative index, a-n

= _________

c) Zero index, a0

= __________

Equations involving indices

a) If ax = ay, then _________

b) If mx

= nx, then ________



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5.2 INDICES AND LAWS OF INDICES

Examples Hit the Buttons

Find the values of the following.

1. 63 = 216

2. 4-2 =2

4

1

=16

1

3. 32

8 = ( 3 8 )2 = 22

= 4

EXERCISE 5.2

a) Evaluate the following

1. 0.44

Solution

2. 32

8

27

Solution

3.

3

8

1

Solution

4.2

1

4

1

Solution

6 ^ 3 =

( 3 shift ^ 8

) =x2

1c

ba 4 =x2


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5.3 SIMPLIFY ALGEBRAIC EXPRESSIONS

Examples Solution

Simplify

1. 2a2 x 3a3

= (2 x 3) x (a2 x a3)= 6 x a2 + 3

Sign : (+) x (+) = (+)

Number : 2 x 3 = 6Unknown/ : a2 x a3 = a5

base

2. 33n 2

3n 3

= 33n 2 - ( n 3)

= 33n 2 n + 3

= 3 2n + 1

am an = a m n

Use the laws of indices to simplify thedivision

EXERCISE 5.3

Simplify the following

1. (34)2n x 3n

Solution

2. 31 2n x 34 x 33n + 2

Solution

3. 121 48 nn

Solution

4. a2 x 2a3 (4a)2

Solution


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5.3n5

2n-32

p

pp

n

Solution

6.22

27

b6a

b18a

Solution

7. 3x 3x 1

Solution

8. 2n + 2 - 2n + 12( 2n 1)

Solution

5.4 EQUATIONS INVOLVING INDICES

Examples Steps / Method

Solve each of the following equations

1. 33x = 8133x = 34

3 x = 4

x = 3

4

Express in the _________________ andcompare the indices

_____________________ the indices

2. 2x . 4x+1 = 642x . 22 (x+1) = 26

x + 2x + 2 = 6

3x = 4

x =3

4

Use ______ as the same base throughout

_______________ the indices using the

laws of indices


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3. 0168 12

xx

022 1432

xx

143 222

xx

443

22

2 xx

3x2

= 4x + 4

3x2- 4x 4 = 0

(3x + 2)(x 2) = 0

x =3

2, x = 2

Use 2 as the same base throughout

______________________of a quadraticequation

_________________________________

EXERCISE 5.4

1. Solve 8x3

= 27

Solution

2. Solvex8 =

4

1

Solution

3. Solve 2x . 3

x = 36

Solution

4. Solve 21

4x = 10

Solution


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5. Solve 14 2793 xx

Solution

6. Solve 53x 25x + 1 =

125

1

Solution

7. Solve427

x =xx

93

13

Solution

8. Solve3

1

5

25125

x

x

Solution


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5.5 PAST YEAR QUESTIONS

1. Solve the equation 122 34 xx (SPM 05)

Solution

2. Solve the equation 684 432 xx (SPM 04)

Solution


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5.6 ASSESSMENT

Answer All the questions Time : 30 minutes

1. Solve the equation 163x = 86x + 1

Solution

2. Solve the equation 32x =

243

1

Solution

3.Solve the equation 8x . 2x = 32

Solution


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4.Solve the equation 093 26

2

xx

Solution

5. Solve the equation28 x =

xx

24

11

Solution

6.Solve the equation 52x 52x 1 = 100

Solution


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ANSWERS

EXERCISE 1.1

1. 0.0256

2. 2.253. 512

4 2

EXERCISE 2.1

1. 39n

5. 12 n

2. 3n+7

6. 2a5

3.np 4

17.

3

23x

4. 3a

8

18. 92n

EQUATION INVOLVING INDICES1. same base, compare

2. 2, Add

3. General form, solving by factorization

EXERCISE 3.1

1. x =2

35. x = 6

2. x =3

26. x = -1

3. x = 2 7. x = -2

4. x = 6.25 8. x = 1

PAST YEAR QUESTIONS

1. x = -1

2. x = 3

ASSESSMENT

1. x =2

54. x = 6, -2

2. x =2

1 5. x = 2

3. x =4

56. x =

2

3


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ADDITIONAL

MATHEMATICS

MODULE 9


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__________________________________________


CONTENTS PAGE

5.0 INDICES AND LOGARITHM

5.1 Conceptual Map 2

5.2 Logarithm and laws of logarithm 3 4

Exercise 5.2

5.3 Finding Logarithm of numbers 4 6

Exercise 5.3

5.4 Changing the base of logarithm 7 - 9

Exercise 5.4.1

Exercise 5.4.2

Exercise 5.4.3

5.5 Equation Involving logarithm 10 11

Exercise 5.5

5.6 Past Year Questions 11 - 12

5.7 Assessment 13

Answers 14


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5.1 CONCEPTUAL MAP

Index

an index

base

Logarithms

If N = ax, then loga N = x

Laws of Logarithms

d) Loga xy = _________________

e) Loga x/y = _________________

f) Loga xm = _________________

Change of base

of logarithms

a) Loga b = __________

b) Loga b = __________

Equations involving

Logarithms

If logaX = logaY, then

__________________



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5.2 LOGARITHM AND LAWS OF LOGARITHM

Find the value of the following

Examples Solution

1. log4641

Let . log464

1= x

Then 4x = 4-3

x = -3

2. log5= 5 Let . log5 5 = x

Then 5x = 21

5

x =2

1

3. log2 x = - 5 Given log2 x = - 5Then x = 2

-5

=52

1 =321

EXERCISE 5.2

Find the value of each of the following

1. log2 128 2 log5 125

3. . log864

14. log10

3

5

2


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5. log100 10 6. log 644

7. log9 x = - 1 8. logx 225

1

5.3 FINDING LOGARITHM OF NUMBERS

Examples Solution

Evaluate the following

1. log3 18 log3 6

= log36

18

= log3 3

= 1

2. log4 2 + log4 8= log4 (2 x 8)

= log4 42

= 2 log4 4

= 2(1)= 2

3. Given that loga 2 = 0.123 and loga 3 = 0.256, calculate the following

a) loga 12 = loga (2 x 2 x 3)= loga 2

2 + loga 3

= 2loga 2 + loga 3

= 2(0.123) + 0.256= 0.502


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b) loga 6 = loga 21

6

=2

1loga (2 x 3)

=2

1[ loga 2 + loga 3]

=2

1[0.123 + 0.256]

= 0.1895

4. Given that logx 2 = a and logx 3 = b, write logx 29

4

xin term of a and b.

Logx 29

4

x= logx 4 logx 29x

= logx 22

- [ logx 9 + logx x2

]= 2 logx2 - logx 3

2 - 2 logx x= 2a - 2logx 3 - 2(1)

= 2a - 2b - 2

EXERCISE 5.3

Simplify each of the following logarithms expression

1. log4 2 + log4 32 2. log2 48 + log23 log29

3. 2 log5 10 + 3 log5 2 log5 32 4. 4logx 2 3logx 4 + logx 6


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5. Express logac

b2in terms ofloga b

and loga c

Solution

6. Given that log2 6 = 2.59 andlog2 5 = 2.32 find the value of

log10 30

Solution

7. Given log3 2 = 0.631 and log3 5 = 1.465, find the values ofa) log3 20

b) log35

4

c) log3 7.5

Solution


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5.4 CHANGING THE BASE OF LOGARITHM

Examples

Find the value of the logarithm

1. log3

5 =3log

5log

10

10

=4771.0

699.0

= 1.465

EXERCISE 5.4.1

Find the value of the logarithm by changing the base of the logarithm

1. Log4 3 2. Log3 10

3. Log2 0.1 4. Log5 0.048

5.Log4

5

1 6.Log2

4

3


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Examples


1. log4 5 =4log

5log

2

2

= 2log2

322.2

2

=2

322.2

= 1.161

EXERCISE 5.4.2

Given that log2 5 = 2.322 and log2 3 = 1.585, find the value of the logarithm

1. Log3 5 2. Log8 15

3. Log9 125 4. Log8 25


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Examples


1. log10 p2 =

16log

log

4

2

4 p

=4log2

log2

4

4 p

=2

2m

= m

EXERCISE 5.4.3

Given that log4 p = m. Express the following in terms of m

1. Logp 4 2. Log8 p

3. Log4p 64 4.Logp

256

1


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5.5 EQUATION INVOLVING LOGARITHMS

1. Solve the equation :

Log 10( x 5 ) = log10 ( x 1 ) + 2

Solution

Log 10( x 5 ) = log10 ( x 1 ) + 2

Log 10( x 5 ) log10 ( x 1 ) = 2

Log 101

5

x

x= 2

1

5

x

x= 10

2

x 5 = 100( x 1)x 5 = 100x 100

99x = 95

x =99

95

2. Solve the equation:

3x

= 52x + 1

Solutionxlog103 = (2x + 1)log105

xlog103 = 2x log105 + log105xlog103 2x log105 = log105

x(log103 2 log105) = log105

x =5log23log

5log

1010

10

x = 0.7591

EXERCISE 5.5

Solve each of the equation

1. 5x

= 27 2. 5x

= 0.8

3. 3x +

= 18 4. 5x

= 4x +


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5. 2x3

x= 5

x +6. 3

x. 4

x += 6

5.6 SPM QUESTIONS

SPM 2005( Paper 1)

1. Solve the equation 1)12(log4log 33 xx [ 3 marks]

SPM 2005( Paper 1)

2. Given that pm log and rm 3log , express

4

27log

mm in term of p and r

[ 4 marks]


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SPM 2004(Paper 1)

3. Given that 5log 2 = m and 5log = p, express 5log 4.9 in term of m and p

[ 4 marks]

SPM 2003(Paper1)

4. Given that 2log T - 4log V = 3 , express T in term of V. Express T in term of V

[ 4 marks]

SPM 2003(Paper1)5. Solve the equation xx 74 12 [ 4 marks]


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5.7 ASSESSMENT

Answer All the questions Time : 30 minutes

1. Solve the equation 4log3log2)48(log 555 x

2. Given x = mk3log and nk2log . Find the value of 24log k in term ofm and n

3. Given that ,4loglog 93 VT express T in terms of V

4. Solve the equation xx 95 12


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ANSWERS

EXERCISE 5.2

1. 7 4 -1.1938 7.9

1

2.2

35.

2

18. 5

3. -2 6. 6

EXERCISE 5.3

1. 3 5. 2 logab loga c

2. 4 6. 4.913. 2 7. a) 2.727 c) 1.834

4. logx2

3b) - 0.203

EXERCISE 5.4.11. 0.7924 3. -3.3222. 2.096 4. -1.887

EXERCISE 5.4.2

1. 1.465 3. 2.19752. 1.302 4. -1.548

EXERCISE 5.4.3

1.m

13.

m1

3

2. m32 4.

m4

EXERCISE 5.5

1. 2.048 4. 20.642. -0.1386 5. 8.827

3. 1.630 6. 0.1632

SPM QUESTIONS

1. x =2

34. T = 8 v

2. 3r 2p + 1 5. 1.677

3. 2p - m -1

ASSESSMENT

1. 5 3. T =v

81

2.nm

31 4. -1.575

Documents

5. Indices & Log