Name____________________________
Hour_________
Unit 5 Notes
Rational Exponents, Exponential, and Logarithmic Functions
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Algebra 2 Unit 5
Learning Goal: Students will be able to use rational exponents and their properties, graph exponential
functions and use decay/growth/interest equations, and evaluate logarithms and use their properties to
solve logarithm problems.
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3
4
5
6
7
8
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Objective in words
Students will be able to use the properties of exponents to simplify expressions.
Students will be able to convert between rational exponents and radical form and
evaluate expressions with rational exponents.
Students will be able to solve equations by taking the nth root.
Students will be able to apply properties of rational exponents to simply expressions
involving variables.
Students will be able to graph exponential growth functions and use the exponential
growth and interest models.
Students will be able to graph exponential decay functions and use the exponential
decay model.
Students will be able to find the inverse equation of a function.
Students will be able to evaluate logarithms and translate between log and exponential
form.
Students will be able to condense and expand logarithms using their properties.
Students will be able to solve exponential and logarithmic equations, using the change
of base formula if necessary.
Score 4
Score 3
Score 2
Score 1
Score 0
0 1 2 3 4
I can use properties of rational exponents to simplify advanced expressions, graph
exponential functions, and apply the properties of logarithms to evaluate, simplify and
solve complex problems. I understand the relationship between exponential functions and
logarithms functions and how they are inverses and can use this to solve problems.
I can use properties of rational exponents to simplify expressions, graph exponential
functions, and apply the properties of logarithms to evaluate, simplify and solve equations
with little mistake. I know that exponential functions and logarithms functions are inverses
of each other.
I can use properties of rational exponents to simplify expressions, graph exponential
functions, and apply the properties of logarithms to evaluate, simplify and solve equations
as long as they are not too complicated.
With help, I can solve these problems as long as they are not too complex.
Even if I have help, I cannot do these problems….YET!
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5.1 Use Properties of Exponents
Properties of Exponents:
Product of Powers Property
Power of a Power Property
Power of a Product Property
a m  a n  ___________
(a m ) n  ___________
(ab) m  ___________
Negative Exponent Property
Zero Exponent Property
a  m  ___________
a 0  ___________
Quotient of Powers Property
am
 ___________
an
a
( ) m  ___________
b
Evaluating Numerical Expressions: Evaluate the expression. Write your answer with
exponents.
45
1. 4 2  4 3
2. (4 2 ) 3
3. 3
4
Power of a Quotient Property
4. (4 2  35 ) 2
5. 4 2
6. 4 0
4
7. ( ) 2
5
4 7
8. 3
4
9. 4 3  4 5
35  315
11.
33
53
12. 3
5
10. 2  2
0
5
Evaluating Algebraic Expressions: Simplify the expression.
13. b 2  b 4
14. (a 2 ) 3
15.
3
z5
z3
16. (a  b) 2
x2  y 
19.


3 y 2  2 x 2 
2
17. x 2
18. (2 2 y 3 ) 5
( x 3 y 3 ) 2
20.
x5 y 6
21. (3a 3b 5 ) 3
SUMMARY
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5.2 Evaluate nth Roots and Use Rational Exponents
Vocabulary:

If bn=a then b is the ____________________________.

An nth root of a is written as
where n is the ______________ of the radical.
The power of a power property applies to rational exponents so the following is true:
etc…
Because a1/2 is a number whose square is a, you can write…

=

=

=
Etc…
Using Rational Exponent Notation:
n
a  ________
 a
3
8  ________
( 16 ) 3  ________=__________
n
m
 n a m  ________
Ex:
Convert from radical form to rational exponent form:
1. 3 12
2.
 10 
7
3
Convert from rational exponent form to radical form:
3. 5
1
4
4. 14
2
5
5
Evaluating expressions:
( 4 16 ) 5  ________
4 5 / 2  ________
Ex:
3
8  ________
813 / 4  ________
Evaluate an expression with rational exponents:
5. 3 125
7.
9.
6
64
8. 16
 8
3
6. 24011 / 4
2
3
2
10. 64
6
2
3
Solving Equations:
Ex 1: Solve 6x3 = 384
Ex 2: Solve (x – 8)5 = 32
11. Solve (x – 2)4 = 1296
Ex 3: Solve 3x6 = 192
12. Solve -2(x + 5)3 = 128
SUMMARY
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5.3 Apply Properties of Rational Exponents
Vocabulary:

A radical with index n is in ______________ _______________ if the radicand has no
perfect nth powers as factors and any denominator has been __________________.

Like radicals are ______________________________.
Properties of Radicals:
 Product Property of Radicals:

Quotient Property of Radicals:

Addition/Subtractions Property of Radicals:
Rationalizing the denominator
 To rationalize a denominator, multiply the numerator and denominator by the radical of
the denominator
1)
2

5
2)
3

3
6
3)
4 2

3 5
4)
1

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Use properties of radicals:
Ex 1:
Ex 2:
5
27  5 9  _____________________ = _________________ = _________
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192
 _____________________ = _________________ = _________
3
3
5
5
Ex 3: 3 9  7 9  ______________________=________________
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Simplify. Write radicals in simplest form: (take out groups based on index)
5.
20  5
6. 3 135
8. 33 5  23 5
5.
45  2 5
7. 54 64  24 8
9. 4 1250  84 32
Simplify expressions involving variables:
10.
5
32x15
12. 73 128a12b11c 5
11.
3
8 x 7 y 3 z 11
13. 4 162 x 5 y 15
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SUMMARY
5.4 Graph Exponential Growth Functions
Vocabulary:

Exponential Function has form: ____________________________.

Exponential Growth Function is __________________________ where
b _________. b is called the ______________________.
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Graph Exponential Functions:
1. y  2 x
Steps to graph:
1st: Make a table.
x
f (x )
2nd: Fill in the table.
3rd: Plot the points.
4th: State the domain and range.
Domain ___________ Range _______________
Asymptote _________
2. y  3 x 1
1st: Make a table.
x
f (x )
2nd: Fill in the table.
3rd: Plot the points.
4th: State the domain and range.
Domain ___________ Range _______________
Asymptote _________
3. y  4 x 1  3
1st: Make a table.
x
f (x )
2nd: Fill in the table.
3rd: Plot the points.
4th: State the domain and range.
Domain ___________ Range _______________
Asymptote _________
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Write A Model For Exponential Growth:
Exponential Growth
Model
y = a (1 + r)t
Compound Interest
y = P (1 + )nt
4) You deposit $2900 in an account that pays 3.5% annual interest. Write a model to represent
the situation for the interest compounded monthly and annually. Then find the balance after 1
year and 5 years.
5) In the last 12 years, an initial population of 38 buffalo on a state park grew by about 7% per
year. Write a model to represent the exponential growth. Then find the number of buffalos in
2005.
6) A car is sold at auction. The owner bought the car in 1984 when its value was $11,000. The
value of the car increased at a rate of 6.9% per year.
a) Write a function that models the value of the car over time.
b) The auction was in 2004. What was the approximate value at this time?
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Continuously Compound Interest:
Continuous Interest
y = P ert

_____ is called the _______________ _______________. According to
n
1

Leonhard Euler, the expression 1   approaches e as n increases.
n

o e  ____________________
7) You deposit $3500 in an account that pays 4% annual interest compound continuously. What
is the balance after 1 year?
8) You deposit $4800 in an account that pays 6.5% annual interest compound continuously.
What is the balance after 3 years?
SUMMARY
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5.5 Graph Exponential Decay Functions
Vocabulary:

Exponential Decay Function is __________________________ where
b _________. b is called the ______________________.
Graph Exponential Decay Functions:
3
1. y   
4
x
Steps to graph:
1st: Make a table.
x
f (x )
-2
-1
0
1
2
2nd: Fill in the table.
3rd: Plot the points.
4th: State the domain and range.
Domain ____________ Range ____________
Asymptote _________
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1
2. y   
2
x 1
4
1st: Make a table.
x
f (x )
-2
-1
0
1
2
2nd: Fill in the table.
3rd: Plot the points.
4th: State the domain and range.
Domain ____________ Range ____________
X- Asymptote _________
Write A Model For Exponential Decay:
Exponential Decay
Model
y = a (1 - r)t
3) A new television costs $1200 in 2006. The value of the television decreases by 21% each
year. Write an exponential decay model. Then find the value of the television in 2013.
4) The number of acres of Ponderosa pine forests decreases by 0.5% annually. In 1963 there
were 41 million acres of Ponderosa pine forests. Write a function that models the number of
pine trees over time. About how many acres of pines where there in 2002?
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SUMMARY
5.6 Finding Inverses of Functions
Vocabulary:
An inverse relation _________________________________________________________
_____________________________________________________________________.
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Finding equations for inverse relations.
Step 1: Write original equation replacing f(x) with y if needed.
Step 2: Switch x and y.
Step 3: Solve for y. This is the inverse.
Step 4: Write the inverse using the notation f-1(x)
1) f(x) = 4x – 1
3) f ( x)  x 2  6
5) f ( x) 
2 x  10
3
2)
4) g ( x)  2 x  7
6) p( x)  x 3  9
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Inverse Functions and composition:
Functions (f and g) are inverses only if:
f ( g ( x))  _________ and g ( f ( x))  _________ . Function g is then called ____
Verify that functions are inverses:
1
4
7) f ( x)  7 x  4 and f 1 ( x)  x 
7
7
8) f ( x)  x 2  1 and f
Find the inverse of the function. Then
graph f and f-1:
9) f ( x)  4 x  2
18
1
( x)  x  1
SUMMARY
5.7 Evaluating Logarithms
Vocabulary:

Logarithm of y with base b is ______________________________________.
o ___________________ if and only if ___________________

A common logarithm is ________________________________.

A natural logarithm is ____________________________.
Rewrite Logarithmic Equations:
Logarithmic Form
Exponential Form
1. log 2 32  5
_______________________
2. log 7 1  0
_______________________ Notice log a 1  0 for any a value
3. log 13 13  1
_______________________ Notice log a a  1 for any a value
4. log 1 2  1
_______________________
2
5. ln e  1
_______________________
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Evaluate Logarithms:
6. log 3 81
_______________________ = ___________
7. log 4 0.25
_______________________= ___________
8. log 1 256
______________________ = ___________
9. log 1 9
________________________ = ___________
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3
Calculating Logarithms with a calculator:
10. log 20  __________________
11. log 34  __________________
12. ln 11  __________________
Find The Inverse: (Remember, replace the function notation with y and then
switch ___ and ___, then solve to y.)
13. f ( x)  log 3 x
14. f ( x)  8 x
15. f ( x)  log 6 x  2
16. f ( x)  7 x 1
2
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Graph A Logarithmic Function:
17.Graph y  2 x and y  log 2 x on the same graph below.
SUMMARY
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5.8 Apply Properties of Logarithms
Properties Of Logarithms
Product Property: log b mn  _____________________________
Quotient Property log b
m
 _____________________________
n
Power Property: log b m n  _____________________________
Approximate Expressions:
Use log 5 4  0.861 and log 5 9  1.365 to evaluate.
1. log 5
4
 _____________________ = __________________ = _________________
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2. log 5 36  _____________________ = __________________ = _________________
Use log 3 2  0.6309 and log 3 8  1.8928 to evaluate.
3. log 3
8

2
4. log 3 16 
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Expand A Logarithmic Expression:
This means to write the expression as the sum and differences of_____________
_________________.
7x2
5. Expand log 3 4 x
6. Expand log 3
y
7. Expand log 4
x
3y
8. Expand ln 6 x 3 y 2
Condense Expressions:
This means to write the expression as a _____________ ___________________.
9. Condense log 2  3 log 3  log 9
10. Condense ln 3  2 ln x  ln y
11. Condense log 6 x  2 log 6 4  log 6 3
12. Condense 6 log x  log 4  log 10  log y
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Use Change-of-Base Formula:
Example:
log 6 11  _________________ = ____________________ (using common logarithms)
Or
log 6 11  _________________ = ____________________ (using natural logarithms)
13. Evaluate log 16 26 both ways to see you get the same answer.
SUMMARY
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5.9 Solve Exponential and Logarithmic Equations
Solving Exponential Equations
(To solve, they must have the same _________________)
1. 5 x  4  25 x 6
2. 8 x 1  32 3 x  2
1
3. 813 x   
3
Solve the equation.
4. 7 3 x  4  49 2 x 1
Solving Exponential Equations
5. Solve 6 x  27
6. Solve 6 x  10  9
25
5 x 6
7. Solve 115 x  33
8. Solve ex+2 = 10
Solving Logarithmic Equations:
9. log 5 (5x  9)  log 5 6 x
Solve by rewriting in exponential form:
10. Solve log 5 (3x  8)  2
11. Solve log 3 ( x  7)  4
(Condense these equations first…)
13. Solve log 2 ( x)  log 2 ( x  2)  3
12. Solve log 5  log( x  1)  2
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14. You deposit $3500 in an account that pays 4% annual interest compound continuously.
After about how many years will your account have $9,000 in it?
15. You deposit $4800 in an account that pays 7% annual interest compounded biannually. After
about how many years will you have $10,000?
SUMMARY
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