2.08 ADD MATH MODULE 08 INDICES AND LOGARITHMS · PDF file1.12.2008 · Solve the...

INDICES AND

ADDITMATHE

MOD

http://mathsmoz

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IONALMATICS

LOGARITHMS

ULE 8

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1

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

Laws of

a) am x an

b) am ÷ an

c) (am)n=

Integer

a) an = a x a x a

n facto

b) Negative index,

c) Zero index, a0=


2

Indices

anindex

base

Logarithms

If N = ax, then loga N = x

Fractional indices

a) a1/n= ___________

b) am/n= ___________

= ___________

indices

= _________

= _________

_________

indices

… x a x a

rs

a-n= _________

__________

Equations involving indices

a) If ax = ay, then _________

b) If mx = nx, then ________

INDICES AND LOGARITHMS


3

5.2 INDICES AND LAWS OF INDICES

Examples Hit the ButtonsFind the values of the following.

1. 63 = 216

2. 4-2 =24

1

=16

1

3. 3

2

8 = ( 3 8 )2 = 22

= 4

EXERCISE 5.2

a) Evaluate the following

1. 0.44

Solution

2. 3

2

8

27

Solution

3.3

8

1

Solution

4.2

1

4

1

Solution

6 ^ 3 =

( 3 shift ^ 8

) =x2

1c

ba 4 =x2



4

5.3 SIMPLIFY ALGEBRAIC EXPRESSIONS

Examples SolutionSimplify

1. 2a2 x 3a3

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

Sign : (+) x (+) = (+)Number : 2 x 3 = 6

Unknown/ : 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



5

5.3n5

2n-32

p

pp

n

Solution

6.22

27

b6a

b18a

Solution

7. 3x – 3

x – 1

Solution

8. 2n + 2 - 2

n + 12( 2n – 1)

Solution

5.4 EQUATIONS INVOLVING INDICES

Examples Steps / MethodSolve each of the following equations

1. 33x = 81

33x = 34

3x = 4

x =3

4

Express in the _________________ andcompare the indices

_____________________ the indices

2. 2x . 4x+1 = 64

2x . 22 (x+1) = 26

x + 2x + 2 = 63x = 4

x =3

4

Use ______ as the same base throughout

_______________ the indices using thelaws of indices



6

3. 0168 12

xx

022143

2

xx

143 222

xx

443 222 xx

3x2 = 4x + 43x2- 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. Solve x8 =4

1

Solution

3. Solve 2x . 3

x = 36

Solution

4. Solve 2

1

4x = 10

Solution



7

5. Solve 14 2793 xx

Solution

6. Solve 53x 25x + 1 =125

1

Solution

7. Solve 427 x =xx 93

13

Solution

8. Solve3

1

5

25125

x

x

Solution



8

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



9

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



10

4.Solve the equation 093 262

xx

Solution

5. Solve the equation 28 x =xx 24

11

Solution

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

Solution



11

ANSWERS

EXERCISE 1.1

1. 0.02562. 2.253. 5124 2

EXERCISE 2.1

1. 39n 5. 12 n

2. 3n+7 6. 2a5

3.np 4

17.

3

23x

4. 3a8

18. 92n

EQUATION INVOLVING INDICES1. same base, compare2. 2, Add3. 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 = -24. x = 6.25 8. x = 1

PAST YEAR QUESTIONS1. x = -12. x = 3

ASSESSMENT

1. x =2

54. x = 6, -2

2. x =2

15. x = 2

3. x =4

56. x =

2

3



12

INDICES AND

ADDITMATHE

MOD

http://mathsmoz


IONALMATICS

LOGARITHMS

ULE 9

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13

__________________________________________


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




5.1 CONCEPTUAL MAP

Index

an

base

Equations involvinLogarithms

If logaX = logaY, then

_________________

ht


14

indexLogarithms

If N = ax, then loga N = x

Laws of Logarithms

d) Loga xy = _________________

e) Loga x/y = _________________

f) Loga xm= _________________

Change of baseof logarithms

a) Loga b = __________

b) Loga b = __________

g

_

INDICES AND LOGARITHMS

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5.2 LOGARITHM AND LAWS OF LOGARITHM

Find the value of the followingExamples Solution

1. log464

1Let . log4

64

1= x

Then 4x = 4-3

x = -3

2. log5= 5 Let . log5 5 = x

Then 5x = 2

1

5

x =2

1

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

=52

1=

32

1

EXERCISE 5.2

Find the value of each of the following

1. log2 128 2 log5 125

3. . log864

14. log10

3

5

2



16

5. log100 10 6. log 644

7. log9 x = -1 8. logx 225

1

5.3 FINDING LOGARITHM OF NUMBERS

Examples SolutionEvaluate 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 22 + loga 3= 2loga 2 + loga 3= 2(0.123) + 0.256= 0.502



17

b) loga 6 = loga 2

1

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 32 - 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



18

5. Express logac

b2

in terms of loga b

and loga c

Solution

6. Given that log2 6 = 2.59 andlog2 5 = 2.32 find the value oflog10 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



19

5.4 CHANGING THE BASE OF LOGARITHM

ExamplesFind 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



20


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



21


1. log10 p2 =16log

log

4

24 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



22

5.5 EQUATION INVOLVING LOGARITHMS

1. Solve the equation :Log 10( x – 5 ) = log10 ( x – 1 ) + 2

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

Log 101

5

x

x= 2

1

5

x

x= 102

x – 5 = 100( x – 1)x – 5 = 100x – 10099x = 95

x =99

95

2. Solve the equation:3x = 52x + 1

Solutionxlog103 = (2x + 1)log105xlog103 = 2x log105 + log105xlog103 – 2x log105 = log105x(log103 – 2 log105) = log105

x =5log23log

5log

1010

10

x = – 0.7591

EXERCISE 5.5

Solve each of the equation1. 5x = 27 2. 5x = 0.8

3. 3x + 1 = 18 4. 5x – 2 = 4x + 1



23

5. 2x3x = 5x + 1 6. 3x . 4x + 1 = 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]



24

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]



25

5.7 ASSESSMENT

Answer All the questions Time : 30 minutes

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

2. Given x = mk 3log and nk 2log . Find the value of 24log k in term of m and n

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

4. Solve the equation xx 95 12



26

ANSWERSEXERCISE 5.2

1. 7 4 -1.1938 7.9

1

2.2

35.

2

18. 5

3. -2 6. 6

EXERCISE 5.31. 3 5. 2 logab – loga c2. 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 5.4.21. 1.465 3. 2.19752. 1.302 4. -1.548

EXERCISE 5.4.3

1.m

13.

m1

3

2. m3

24.

m

4

EXERCISE 5.51. 2.048 4. 20.642. -0.1386 5. 8.8273. 1.630 6. 0.1632

SPM QUESTIONS

1. x =2

34. T = 8 v

2. 3r – 2p + 1 5. 1.6773. 2p - m -1

ASSESSMENT

1. 5 3. T =v

81

2.nm

31 4. -1.575



2.08 ADD MATH MODULE 08 INDICES AND LOGARITHMS · PDF file1.12.2008 · Solve the...

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