Chptr5 Indices

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ADDITIONAL MATHEMATICS FORM 4 NAME: _____________________ CHAPTER 5: INDICES & LOGARITHMS FORM:__________

1.1 Find the value of numbers given in the form of:a) integer indicesb) fractional indices

1.2 Use laws of indices to find the value of numbers in index form that are multiplied, divided or raised to a power.1.3 Use laws of indices to simplify algebraic expressions.

Laws of indices

1. Simplify23

312

3

33

n

nn

Ans:

9

1

2. Simplify1

3

44

416+

nn

n

Ans:

241+n

3. Simplifyn

nn

32

3123

3

26+

+ Ans: 8 4. Simplifymm

mm

321

1

52

1025+

+

Ans:

5

4

5. Simplify ( )223325 y x y x Ans:

y x8

56. Simplify

2

1

4

2

q

p Ans: p

q2

1

6. Zero index =>7. Negative index =>8. Fractional index

i)ii)

1.2.3.4.5.

Date: ________ Day: _________


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7. Express nnn 555 12 ++ in the simplest formAns:

)5(19 n

8. Given that nnnn k 552555 12 = + , where k isa constant, find the value of k. Ans: k=19

9. Solve the following simultaneous equations

2793 2 = y x and81

42 = y x Ans: x= -1, y=110. Solve the following simultaneous equations

1497 = y x and625

1255 3 = y x Ans: x= -2, y=1

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2.1 Express equation in index form to logarithm form and vice versa2.2 Find logarithm of a number 2.3 Find logarithm of numbers by using laws of logarithms.2.4 Simplify logarithmic expressions to the simplest form.

Logarithms Law Of Logarithms

1. Convert the following to logarithm forma) 164 2 = b)

81

2 3 =

c) 100010 3 = d) y x =2

2. Convert the following to index forma) 327log 3 = b) 1

41

log 4 =

c) 3001.0log 10 = d) x y =3log

3. Given that 501.02log =a and 794.03log =a , find the value of a) 6log a Ans: 1.295 b)

32log a Ans: - 0.293

c) 9log a Ans:1.588 d) aa

a

2log9log

Ans: 1.512

3

Date: ________ Day: _________

1. x y ya a x == log

2. 1log =aa3. 01log =a4. xa xa =

log

5. Logarithm of a negative number is undefined6. Logarithm of zero is undefined

1. y x xy aaa logloglog +=

2. y x y x

aaa logloglog =

3. xn x an

a loglog =


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4. Given that 431.02log 5 = and 682.03log 5 = , find the value of a) 6log 5 Ans: 1.113 b) 5.1log 5 Ans:

0.251

c) 3log 5 Ans:0.341 d) 10log

9log

5

5 Ans: 0.953

5. Given that 257.03log =a and 524.05log =a , find the value of a) 53log a Ans: 0.519

b)aa

a

3log25log

Ans: 0.834

c) ( )25log aa Ans: 2.524 d)

aa 5

9log Ans: -1.01

6. Evaluate each of the following without using a calculator.a) 32log2log 44 + Ans: 3 b) 6log23log24log 555 + Ans: 0

c) 27log3log 99 + Ans: 2 d)34

log212log4log 333 + Ans: 3

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7. Given that r xa =log and s ya =log , express each of the following in terms of r and s.a) 243log a y xa Ans:

243 ++ sr b) 42log

ya

xa Ans:

sr 21

2

1

8. Given that ma =3log and k a =5log , express

2

135log aa in terms of m and k. Ans: 3m+k-2

9. Given that r m 2= and t n 2= , express

8

log23

2nm

in terms of r and t. Ans: 3r+2t-3

10. Given that r m 2= and t n 2= , express

16

log2

2nm

in terms of r and t. Ans: 2r+t-4

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3.1 Find the logarithms of a number by changing the base of the logarithms to suitable base.3.2 Solve problems involving the change of base and laws of logarithms.

Change Of Base Of Logarithms

1. Find the values of the following by changing the base to 10.a) 10log 5 Ans: 1.431 b) 5log 25 Ans:

21

c) 17log 6.0 Ans:-5.546

d) 8.8log 2 Ans: 3.138

2. Given that q p =8log , express the following in terms of qa) 28log p Ans:

q

2

b) p8log Ans:

q21

c) p8log 8 Ans:

q

q 1+d) p2log Ans:

q

3

7

a

bb

c

ca log

loglog =

Date: ________ Day: _________


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4.1 Solve equations involving indices.4.2 Solve equations involving logarithms.

Index Equations Logarithmic Equations

INDICES

Solve each of the following equations1. x x = 1279 Ans:

5

32. y y = 2636 Ans:

32

3. 2439 23 = x Ans:4

1

4. 633 1 = + x x Ans: 1

5. 7233 2 +=+ x x Ans: 2 6. 1622 12 += ++ x x Ans:3

8

If thenIf thenIf thenIf then

If , thenIf , then

Date: ________ Day: _________


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7.16714 2

5

=k Ans: 4 8. 162 23

= x Ans: 4

9.161

2 25

=

p Ans: 410. 2523 += x x x Ans: 17.6549

11. 21 52 + = x x Ans:4.2694

12. 2553 += x Ans: - 6.301

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LOGARITHMS

Solve each of the following equations.

1. ( ) 41loglog 22 = x x Ans :

1716

2. ( ) y y += 1log23log 33 Ans:

4

3

3. 3log3log 33 =+ x x Ans: 3 4. ( ) mm 55 log114log = Ans:

45

5. 9log23

3log x x = Ans: 9 6.54log8log =+ x x Ans: 2

7.23

9log3log 22 =+ ++ y y Ans: 79 8. 35

8log4log 11 =+ ++ y y Ans: 63

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9. 82 3log = x Ans: 27 10. ( )51

5 4log 2 = y Ans:

29

11. 93 2log = y Ans: 4 12. ( ) 255 2log3 = p Ans: 11

Express y in terms of x in each of the following1. 3loglog 22 =+ y x Ans:

x

8

2. ( ) 2loglog 33 = y x y Ans:

109 x

3. ( ) 2loglog 22 = x x y Ans: x y 5=

4. 3loglog 93 =+ x y Ans:

x

27

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5. 1loglog 24 =+ y x Ans:

x2

6. x y 24 log2log = Ans:

2

16 x

7. Given that pm =2log , express the following in terms of pa) 28 4log mm Ans:

p p++3 22

b) mm 8log 4 Ans: p p

246

+

+

8. Given that t p =3log , express the following in terms of t

a) p9log 27 Ans:3

2 t +b) p81log 3 Ans: 2

8 t +

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