Download - Radical and exponents (2)
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Radical and exponents
Chapter 3
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Exponential notation
represent as to the th power .
Exponent (integers)Base
(real number)
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General case (n is any positive integers)
Special cases
Zero and negative exponent(where a c ≠ 0)
Example
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Law of exponentsLaw Example
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Theorem on negative exponents
Prove:
Prove:
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Example :simplifying negative exponents
(1)
8
6
682
23242
234
9
3
)()()3
1(
)3
1(
x
y
yx
yx
yx
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Principal nth root Where n=positive integer greater than 1
= real number
Value for Value for
= positive real number b
Such that =negative real number b
Such that
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Properties of RADICAL
radicandindex
Radical sign
PROPERTY EXAMPLE
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Example:combining radicals
12 5
125
125
32
41
41
3 2
4
1
1
32
α
α
αα
α
α
α
α
)(
Question:
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Law of radicals
law example
WARNING!
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Example:Removing factors from radicals
Question:
aba
aba
aba
aba
baba
baba
23
2)3(
)2()3(
)2)(3(
3.2.3
63
23
223
223
462
532
532
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Rationalizing a denominator
Factor in denominator Multiply numerator and denominator by
Resulting factor
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How do you know when a radical problem is done?
(1) No radicals can be simplified.Example:
(2) There are no fractions in the radical.Example:
(3) There are no radicals in the denominator.Example:
8
1
4
1
5
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Example :Rationalizing denominators
(1)
=
(2)
=
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Definition of rational exponents
m/n = rational number n = positive integer greater than 1 = real number, then
(1)
(2)
(3)
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Example: Simplifying rational powers
(1)
23
2
4
2
6
46
.
yx
yx
yx
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Look at these examples and try to find the pattern…
How do you simplify variables in the radical?
x7
1x x2x x3x x x4 2x x5 2x x x6 3x x
What is the answer to ? x7
7 3x x x
As a general rule, divide the exponent by two. The remainder stays in the
radical.
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LOGARITHMS
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Definition of
• The logarithms of with base is defined by:
if and only if
For every and every real number .
base
exponent
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Illustration
Logarithmic form Exponential form
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• The logarithmic function with base is one-to-one. Thus, the following equivalent conditions are satisfied for positive real number and .
(1) If , then . (2) If , then .
1x 2x
21 xx 21 xx
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Example :Solving a logarithms equation.
Since is a true statement, then
Check..
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• Definition of common logarithm: for every
• Defition of natural logarithm: for every
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Properties of logarithmsLogarithms with base Common logarithms Natural logarithms
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(a) log28=
=
Power to which you need to raise 2 in order to get 8
3 ( Since 23 = 8 )
(b) log41=
=
Power to which you need to raise 4 in order to get 1
0 ( Since 40 = 1 )
(c) log1010,000=
=
Power to which you need to raise 10 in order to get 10,000
4 ( Since 104 = 10,000 )
(d) log101/100=
=
Power to which you need to raise 10 in order to get 1/100
2 ( Since 10-2 = 1/100 )
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Laws of logarithmsCommon logarithms Natural logarithms
WARNING!!
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Example:Application law of logarithm
• log abc² d 3
= log (abc²) − log d 3 = log a + log b + log c² − log d 3 = log a + log b + 2 log c − 3 log d
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Change of base formula
• If and if and are positive real number, then
WARNING!!
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Special change of base formula
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Example
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Example:
Solve Solution :
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QUESTION
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Question 1
Simplify:
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Question 2
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Question 3Simplifying:
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Question 4Simplifying:
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Question 5:
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Question 6
• Solve logb(x2) = logb(2x – 1).
x2 = 2x – 1 x2 – 2x + 1 = 0
(x – 1)(x – 1) = 0
Then the solution is x = 1.
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Question 7
• Solve ln( ex ) = ln( e3 ) + ln( e5 ) ln( ex ) = ln( e3 ) + ln( e5 ) ln( ex ) = ln(( e3 )( e5 )) ln( ex ) = ln( e3 + 5 ) ln( ex ) = ln( e8 )Comparing the arguments: ex = e8 x = 8
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Question 8
Solve log2(x) + log2(x – 2) = 3
log2(x) + log2(x – 2) = 3 log2((x)(x – 2)) = 3 log2(x2 – 2x) = 3
23 = x2 – 2x 8 = x2 – 2x 0 = x2 – 2x – 8 0 = (x – 4)(x + 2) x = 4, –2
Since logs cannot have zero or negative arguments, then the solution to the original equation cannot be x = –2.
Solution: x=4
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Question 9
• SOLUTION:
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Therefore or
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Question 10
If log10 5 + log10 (5x + 1) = log10 (x + 5) + 1,
SOLUTION :
log10 5 + log10 (5x + 1) = log10 (x + 5) + 1
log10 5 + log10 (5x + 1) = log10 (x + 5) + log10 10
log10 [5 (5x + 1)] = log10 [10(x + 5)]
5(5x + 1) = 10(x + 5) 5x + 1 = 2x + 10 3x = 9 x = 3
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Thank you…Prepared by:
Nurul Atiyah binti Ripin (D20111048011)
Irma Naziela binti Rosli (D20111048007)