Exponents, Surds and Logarithms(Gr 12)...Exponents, Surds and Logarithms(Gr 12) Grade 11 CAPS...

Exponents, Surds

and

Logarithms(Gr 12)

Grade 11

CAPS

Mathematics

Series

Outcomes for this Topic

In this Topic you will:

• Revise the exponential notation and review the index laws.

Unit 1.

• Simplify expressions involving rational exponents.

Unit 2.

• Simplify expressions involving surds.

Unit 3.

• Revise the logarithmic notation and logarithm laws.

(mostly done in Grade 12) Unit 4.

Grade 11

CAPS

Mathematics

Series

The

Exponential

Notation and

Index Laws

Unit 1

The Exponential Notation

42 2 2 2 2

We know that :

... (to factors of , , )na a a a a n a n a

nr a

4or 2 is the product of 4 factors of 2.

Exponent

Base

Power

What if exponents are not positive integers?

0

21. 3 1a

0 0

If variable bases are non-zero and is a positive integer then:

1. 1 (0 is undefined)

12. n

n

n

a

aa

2

2

1 12. 3

3 9

Examples

Index Law 1 (Multiplication)

Law 1: n m n ma a a 4 5 91. a a a

1 22. 2 .2 2n n

2 33. 2 . 2 2 8

3 2 5

4. 3 . 3 3 243

Examples

Index Law 2 (Division)

Law 2: n m n ma a a 5

5 3 2

31.

aa a

a

66 8 2

8 2

5 1 12. 5 5

5 5 25

2 1

2

23.

10 5 5

ab a b b

a b a

Examples

Index Law 3 (Exponentiation)

Law 3: m

n nma a

3

2 2 3 61. a a a

2

5 5 2 102. x x x

Examples

Index Law 4

Law 4: ( )m m mab a b

2

3 4 3 2 4 2 6 8 81. 2 2 2 64a a a a

3

3 2 3 3 2 3 9 62. 2 3 2 3 2 3

Examples

Index Law 5

Law 5:

m m

m

a a

b b

3

3 3 3 9

4 4 3 12

2 2 21.

3 3 3

2 2

2 2

2 2 42.

x x x

Examples

Example 1: Applying Exponent Laws

• Simplification using the exponent laws.

6 3

24

6 .91.

118 .4

x x

xx

36 2

4 22 2

2.3 . 3

2.3 . 2

xx

x x

6 6 6

4 8 4 2

2 .3 .3

2 .3 .2

x x x

x x x

6 4 4 2 6 6 82 .3x x x x x x 4 4 42 .3 16 3x xor

Example 2: Applying Exponent Laws

1 1

1 11

2 42.

2 2

n n

n nn n

2

2

1 1

2 2

2 2

22

n n

nn n

2 21 1 2 22n n n n n 2 1

24

• Simplification using the exponent laws.

Multiply with the reciprocal

Tutorial 1: Simplify Expressions using

the Exponent Laws

3 2 3

2

3 2

2 2

4

21

1

Simplify the following expressions:

2 .81.

14 .

4

12 22.

8 3

18 2.33.

2

n n

n

x x

x x

x x

x

PAUSE Unit

• Do Tutorial 1

• Then View Solutions

Tutorial 1 Problem 1: Suggested Solution

3 2 3

2

3 2

2 .81.

14 .

4

n n

n

3 2 3 9

6 4 4

2 . 2

2 . 2

n n

n

3 2 3 9 6 4 42 n n n 2


2 2

4

12 22.

8 3

x x

x x

22 2

3 4

3 2 2

2 3

xx

xx

2 4 2 3 2 42 .3x x x x x 2 2 9

2 .34

2 4 2 2

3 4

2 .3 .2

2 .3

x x x

x x


2

1

1

18 2.33.

2

x x

x

2 1 2 2 22 .3x x x x 72

2 2 2 2

1

2 3 2 3

2

x x x

x

3 22 3

Grade 11 CAPS

Mathematics

Series

Unit 2

Rational

Exponents

What is a rational exponent?

Rational Number is a real number

which can be written in the form

where , and 0m

m n nn

We will in this part consider powers

where the exponent is a rational number.

We will consider expressions like m

na

Equivalent Notations

1

( , 2, )nna a n n 1

3 35 5 (From right to left)

1

554 4 (From left to right)

Equivalent Notations for negative

rational exponents

1 1

1 ( , 2, )nn na a a n n

1

3 13 31

5 5 (From left to right)5

1

5 1 551

= 4 =4 (From right to left)4

Another Equivalent Notation

2

3 232 2 (From left to right)

( 0, ; , 2) m

n mna a r a n m n

5

54 43 3 (From right to left)

• Apply the exponential laws.

• Factorize all bases into prime factors.

21

341. 81 27

1 2

4 34 33 3

1 23 3

1 13

3

1 1

6 42. 125 25

1 1

3 26 45 5

1 1

2 25 5

05 1

Simplification (Without a calculator) of

single term exponential expressions

• Without variables – simplify each term and add.

2 3 2

3 4 33 4 35 2 2

2 23

3 341. 125 16 8

2 3 25 2 2 1 3

25 8 324 4

Example 1: Simplification of polynomial

exponential expressions


• With variables – factorize.

12

2

9 32.

3

x

x

x

2 12

2

3 3 .3

3 .3

x

x

x

13 3 .

3

3 .9

x x

x

13 1

3

3 .9

x

x

2

3

9

2 1 2

3 9 27





2

1

3.2 4.23.

2 2

m m

m m

2 3 16

2 1 2

m

m

1313

1





22 2 64.

2 2

x x

x

2 2 2 3

2 2

x x

x

2

2

Note: Replace 2 with

2 2 6

6

2 3

2 2 2 3

x

x x

x x

a

a a

a a

2 3x





11 12 6.2

5. 5.4

x x x

x

1

2

2 2 3

5.2

x x

x

1

2

5.2

5.2

x x

x

1

2 x x1 1

22



Tutorial 2: Working with exponents

PAUSE Unit

• Do Tutorial 2


3223

1 02

22

3

0,125

5 415

327

1. Simplify without calculators:

4 (a)

9

(b)

2 2

4

2 1

23

2 2

1

2.

12 2

8 3

4 2

2 2 2

5 .2

10 10 .2

x x

x x

x x

x x x

a a

a a

Simplify:

(a)

(b)

(c)

32230,125

1. a Simplify, without using a calculator:

4

9

Tutorial 2 Problem 1(a): Suggested solution

3 22 32 32 1

3 2

3 22 1

3 2

28 82 4

27 27

Tutorial 2 Problem 1(b): Suggested solution

1 02

22

3

5 415

327

1.(b) Simplify (No calculators) :

2 22

3 3

1 1 1

3 .5 3 53

2 2 2 2

1 1 1

3 .5 3 3 5

2 1

1 5

3 .5 9

2 2

4

12 2

8 3

x x

x x

2. a Simplify:

Tutorial 2 Problem 2(a): Suggested solution

2

2 2

3 4

2 .3 2

2 3

xx

x x

2 4 2 2

3 4

2 .3 .2

2 .3

x x x

x x

2 4 2 3 2 42 .3x x x x x

2 2 92 .3

4

2 1

23

4 2

2 2 2

x x

x x x

2. b Simplify:

Tutorial 2 Problem 2(b): Suggested solution

2 2 1

2 2 3

2 2

2 2

x x

x x

2

2

2 1 2

2 1 8

x

x

1 1

7 7

Tutorial 2 Problem 2(c): Suggested solution

2 2

1

5 5 .2 .2

5.2 5.2 .2

a a

a a

2 2

1

5 .2

10 10 .2

a a

a a

2. c Simplify:

2 2

1 1

5 5 .2 .2

5 .2 5 .2 .2

a a

a a a a

2 2

1 1

5 5 .2 .2

5 .2 (1 5 .2 .2)

a a

a a

2 25 2

11

5

4

4 5 1254 25 4 5

5

Grade 11

CAPS

Mathematics

Series

Unit 3

Surds

We Define

1

Thus n na a

,

.

If then

where and

nn a x x a

n a

0 0.n a x

NOTE

If is even, we must have and

Definition of Surds

11

3 33 31. 8 8 2 2

1

1 5 55 52. 32 32 2 2

11

4 44 43. 81 81 3 3

1

1 2 22 24. 16 16 16 4

16 has no meaning (Imaginary numbers)

2Note: a a

Examples directly from the Definition

Use definition

to check!

Property 1: n n na b ab

Applications of the Multiplication Property

1. 2 3 6 (Left to right)

2. 12 4 3 2 3 (Right to left)

Multiplication Property for Surds

3 63. 10 10 (From left to right)

Property 2: m n mna a

3 36 24. 16 16 4 (From right to left)

Composition Property for Surds

Applications of the Composition Property

Property 3: n

m nm a a

Application of the Exponent Property

3 3 344 45. 2 2 8a a a

Exponent Property for Surds

Property 4: n

nn

a a

bb

Applications of the Division Property

2 16. (From left to right)

36

9 9 37. (From right to left)

4 24

Division Property for Surds

1. 2 8 4 32 3 50

2 4 2 4 16 2 3 25 2

4 2 16 2 15 2

3 2

Example 1: Simplification of surd expressions

without using a calculator

2. 50 18 32

5 2 3 2 4 2

5 2 7 2

35 2 70 3 3 3Note: also a a a a a a a



2 2

3. 2 3 2 3

4 4 3 3 4 4 3 3

14



1 14.

2 8

1 1

2 2 2

2 1

2 2

3

2 2

3 2

2 2 2

3 2

4



Moving the surd from the

denominator to the numerator

1 15.

3 2 3 2

3 2 3 2

3 2 3 2

4

3 4

4



1

1 16.

8 2 8 3 2

11 1

3 2 5 2

15 3

15 2

15 2

2



5 11 127.

45 33

5 11 2 3

3 5 3 11

10

3 5 10 5

15

2 5

3



12 32

Simplify, without a calculator:

1.

Tutorial 3: Surds

PAUSE Unit

• Do Tutorial 3


1

1 1

12 3 12 3 3

3.

3 12 27

7 3 75

2.

12 32

Simplify, without a calculator:

1.

Tutorial 3 Example 1: Suggested solution

Easier option?

12 2 12 3 3

15 2 36

15 2 6 27


6 3 3 3

7 3 5 3

3 12 27

7 3 75

2. Simplify without a calculator

3 3 1

412 3


1

1 1

12 3 12 3 3

3. Simplify without calculator

1

1 1

2 3 3 2 3 3 3

1

1 1

3 3 5 3

1

5 3

15 3

15 3

2

Grade 11

CAPS

Mathematics

Series

Unit 4

Logarithms

(mostly for

Grade 12)

If then logy

ax a y x

Exponential form Logarithmic form

log log (zero and negative number)

Both undefined

a a

Example

Definition Logarithms

22 or logyx y x

Note:

0

1 (Trivial)

0

a

a

x

•

•

•

21. From log 8 x

3x

22. From log 5p 5to 2p

32p

3. From log 25 2b 2

2 1to 25

5b

1

5b

3to 2 8 2x

Basis the same

Exponents the same

Exponents the same

Basis the same

Changing from Exponential form to Logarithmic

form and visa verse

log 1 if 0 and 1 m m m m• 1m m

log 1 0 if 0 and 1t t t• 0 1t

10 log logx x•

No base indicated That the base is 10.

log100 2

log 0,01 2

log1 0,• log10 1,

1 log 0,1 log10 1,•

General Remarks

log log log a a aAB A BLaw 1:

log log log a a aA A B

BLaw 2:

log logr

a aP r P Law 3:

loglog

log s

a

s

PP

aLaw 4:

Let ,A B

Logarithmic Laws

Change Basis

log log log a a aAB A BLaw 1:

log5 log 2 log10 1

log 200 log 2 log100 log 2 2

Applications of Logarithmic Law 1

log log loga a a

AA B

B L : aw 2

2 3log 2 log3 log 6 log log1 0

6


log logr

a aP r P Law 3: 2 35 2

2log5 3log 2 log 2 log log1000 32


120 3 log120 log3x x

log 3 log120x

log120

4.35 Use 78 l

Calog

c3

ulatorx

loglog

log s

a

s

PP

aLaw 4:

25

log125 3log5 3log 125

log 25 2log5 2

2 24

2 2

log 8 3log 2 3log 8

log 4 2log 2 2

Application of Logarithmic Law 4

log x

a a x

4

3log 3 44

2 2log 16 log 2 4

3

5 5

1log log 5 3

125

4

4

1 1

2 2

1log 2 log 4

2

log loglog

log log

xx

a

a x aa x

a a

Useful Logarithm Hint

2 3 71. log 8 log 27 log 7 3 3 1 5

2. 2 log 60 2log 2 2log3 2

2 2

60log

2 3

3600log

36 log100 2

0,23. log 125

log125

log 0,2

3

1

log5

log5 3

Simplification of Logarithmic expressions


8 3 3 814. log 16 2log log 1 log 0, 25

9

1log2 2 log 34log 2 403log 2 log 3 log8

2 log 244

3 3log 2

2

4 4 45. 5log 2 log 0,125 2log 8 5

4 3 6

2log

2 2 4log 4 1

log9 log 46.

log8 log 27

2log3 2log 2

3log 2 3log3

2 log3 log 2

3 log3 log 2

2

3

Simplification of Logarithmic expressions


5 7 12

3

1.

3 2log 2log log12

4 5

log 125 log 49 - 2 log 144

log16 - log 9

2log 2 log 3

log 125 log 25

3log 5

x x

x

Evaluate, without a calculator:

(a)

(b)

(c)

(d)

2. log 2 , log3 , log 7 .

, .

log36

log5

12log

7

a b c

a b c

Given

Express the following in terms

of and

(a)

(b)

(c)

Tutorial 4: LogarithmsPAUSE UnitDo Tutorial 4

Then View Solutions

Suggested solutions

Tutorial 4: Logarithms

1. Evaluate, without a calculator:

3 2 3 4 1 1 (a) log 2log log12 log log 2

4 5 4 25 12 100

5 7 12

3log5 2log 7 4log12(b) log 125 log 49 2log 144 3 2 4 1

log5 log 7 log12

2 2log 2 log3log16 log9 4log 2 2log3(c) 2

2log 2 log3 2log 2 log3 2log 2 log3

3

2 73log 5 log 5 log 5

log 125 log 25 7 2 143 3(d) 3 3 3 3 93log 5 log 5 log 52 2

x x xx x

xx x

2. log 2 , log 3 log 7 .

, .

a b c

a b c

Given and

Express the following in terms of and

Suggested solutions

Tutorial 4: Logarithms (continued)

2 2log36 log 2 3 2log 2 2log3 2 2a b (a)

10log5 log log10 log 2 1

2 (b) a

212 1 1log log12 log7 log 2 .3 log7

7 2 2

12log 2 log3 log7

2

12

2 2 2

(c)

b c

a b c a

End of the Topic Slides on Exponents,

Surds and Logarithms

REMEMBER!

•Consult text-books and past exam papers and memos

for additional examples.

•Attempt as many as possible other similar examples on

your own.

•Compare your methods with those that were discussed

in these Topic slides.

•Repeat this procedure until you are confident.

•Do not forget: Practice makes perfect!

Exponents, Surds and Logarithms(Gr 12)...Exponents, Surds and Logarithms(Gr 12) Grade 11 CAPS...

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