Post on 17-Jul-2020
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
Tutorial 1 Problem 2: Suggested Solution
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
Tutorial 1 Problem 3: Suggested Solution
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
• Without variables – simplify each term and add.
• 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
Example 2: Simplification of polynomial
exponential expressions
• Without variables – simplify each term and add.
• With variables – factorize.
2
1
3.2 4.23.
2 2
m m
m m
2 3 16
2 1 2
m
m
1313
1
Example 3: Simplification of polynomial
exponential expressions
• Without variables – simplify each term and add.
• With variables – factorize.
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
Example 4: Simplification of polynomial
exponential expressions
• Without variables – simplify each term and add.
• With variables – factorize.
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
Example 5: Simplification of polynomial
exponential expressions
Tutorial 2: Working with exponents
PAUSE Unit
• Do Tutorial 2
• Then View Solutions
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
Example 2: Simplification of surd expressions
without using a calculator
2 2
3. 2 3 2 3
4 4 3 3 4 4 3 3
14
Example 3: Simplification of surd expressions
without using a calculator
1 14.
2 8
1 1
2 2 2
2 1
2 2
3
2 2
3 2
2 2 2
3 2
4
Example 4: Simplification of surd expressions
without using a calculator
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
Example 5: Simplification of surd expressions
without using a calculator
1
1 16.
8 2 8 3 2
11 1
3 2 5 2
15 3
15 2
15 2
2
Example 6: Simplification of surd expressions
without using a calculator
5 11 127.
45 33
5 11 2 3
3 5 3 11
10
3 5 10 5
15
2 5
3
Example 7: Simplification of surd expressions
without using a calculator
12 32
Simplify, without a calculator:
1.
Tutorial 3: Surds
PAUSE Unit
• Do Tutorial 3
• Then View Solutions
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
Tutorial 3 Example 2: Suggested solution
6 3 3 3
7 3 5 3
3 12 27
7 3 75
2. Simplify without a calculator
3 3 1
412 3
Tutorial 3 Example 3: Suggested solution
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
Applications of Logarithmic Law 2
log logr
a aP r P Law 3: 2 35 2
2log5 3log 2 log 2 log log1000 32
Applications of Logarithmic Law 3
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
without using a calculator
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
without using a calculator
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!