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Honors Advanced Algebra
Chapter 8 – Exponential and Logarithmic
Functions and Relations
Target Goals
By the end of this chapter, you should be able to…
•
•
•
•
•
Graph exponential growth functions. (8.1)
_____ got it
_____needs work
_____ no clue
Graph exponential decay functions. (8.1)
_____ got it
_____needs work
_____ no clue
Solve exponential equations. (8.2)
_____ got it
_____needs work
_____ no clue
Evaluate logarithmic expressions. (8.3)
_____ got it
_____needs work
_____ no clue
Solve logarithmic equations. (8.4)
_____ got it
_____needs work
_____ no clue
•
Simplify and evaluate expressions using the properties of logarithms. (8.5)
_____ got it
_____needs work
_____ no clue
•
Solve logarithmic equations using the properties of logarithms. (8.5)
_____ got it
_____needs work
_____ no clue
•
Solve exponential equations using common logarithms. (8.6)
_____ got it
_____needs work
_____ no clue
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Evaluate logarithmic expressions using the change of base formula. (8.6)
_____ got it
_____needs work
_____ no clue
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Evaluate expressions involving the natural base and natural logarithm. (8.7)
_____ got it
_____needs work
_____ no clue
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Solve exponential equations using natural logarithms. (8.7)
_____ got it
_____needs work
_____ no clue
Honors Advanced Algebra
Chapter 8 - Exponential and Logarithmic
Functions and Relations
Assignment Guide
8.1 – Graphing Exponential Functions
Target Goals: Graph exponential growth functions
Graph exponential decay functions
HW #1
Worksheet #1
8.2 – Solving Exponential Equations
Target Goal: Solve exponential equations
HW #2
pg 488 #1-4, 6, 9-14, 20-22, 33-35
8.3 – Logarithms and Logarithmic Functions
Target Goal: Evaluate logarithmic expressions
HW #3
Worksheet #3
8.4 – Solving Logarithmic Equations
Target Goal: Solve logarithmic equations
HW #4
pg 504 #1-3, 8-19
8.5 – Properties of Logarithms
Target Goals: Simplify and evaluate expressions using the properties of logarithms
Solve logarithmic equations using the properties of logarithms
HW #5
pg 512 #8-11, 23-26, 37-49 odd
8.6 – Common Logarithms
Target Goals: Solve exponential equations using common logarithms
Evaluate logarithmic expressions using the change of base formula
HW #6
Worksheet #6
8.7 – Base e and Natural Logarithms
Target Goals: Evaluate expressions involving the natural base and natural logarithm
Solve exponential equations using natural logarithms
HW #7
pg 529 #1-4, 9-14, 21-27 odd, 35-39 odd, 47, 53, 55
Chapter 8 – Review
HW #8
pg 542 #11, 18-21, 25-28, 31-32, 35-37, 44-47, 49-52, 55, 56-58, 62
Chapter 8 – Review
HW #9
Practice Test
Tentative Chapter 8 Test Date: _______________________
Honors Advanced Algebra
S8.1 Graphing Exponential Functions
Notes
Target Goals:
Name: ________________________
Date: _________________________
Period: ________
Graph exponential growth functions.
Graph exponential decay functions.
A function like f (x ) = b x , where the base is a constant and the exponent contains the independent variable, is
an exponential function. Examples: 4 x , 10 2 x − 3 , and 1000e0.08t
GRAPHING FUNDAMENTAL EXPONENTIAL FUNCTIONS
The graph of the fundamental exponential function f (x ) = b x is continuous and asymptotic in one extreme of
the domain. When b > 1 , the function is increasing and known as an exponential growth function. The function
1

f (x ) = b x contains key points (0,1) , (1,b ), and  −1,  which are plotted to assist graphing the function by

b
hand.
Graph the exponential growth function using three points.
Ex 2. f (x ) = 4 x
Ex 1. y = 2 x
x
y
x
y
When b < 1 , the function is decreasing and known as an exponential decay function.
Graph the exponential decay function using three points.
 1
Ex 3. y =  
 3
x
y
x
 4
Ex 4. f (x ) =  
 5
x
y
x
Ex 5. Which of the following is the graph of y = 2 x − 4 ?
Ex 6. Which of the following is the graph of y = 4 x− 2 + 3 ?
Assignment #1: Worksheet #1
Honors Advanced Algebra
S8.2 Solving Exponential Equations
Notes
Name: ________________________
Date: _________________________
Period: ________
WarmUp:
Rewrite each number as a power with the smallest base possible. Negative exponents are acceptable.
1
1. 9
2. 8
3. 0.1
4.
49
Target Goal:
Solve exponential equations.
In an exponential equation, variables occur as exponents.
PROPERTY OF EQUALITY FOR EXPONENTIAL FUNCTIONS
Let b > 0 and b ≠ 1 . b x = b y if and only if x = y .
Example: If 3x = 35 , then x = 5 . If x = 5 , then 3x = 35 .
The Equality Property can be used to solve exponential equations if the exponential expressions can be written
with the same (common) base.
Solve the exponential equation using the Equality Property.
Ex 1. 3x = 9 4
Ex 2. 8 5 x = 4 2 x −1
Ex 3. 5 2 x − 3 =
1
25
COMPOUND INTEREST
You can calculate compound interest using the exponential function:
r

A = P 1 + 

n
nt
where A is the amount in the account after t years, P is the principal amount invested, r is the annual
interest rate, and n is the number of compounding periods each year.
Solve the compound interest problem.
Ex 4. An investment account pays 4.2% annual interest compounded monthly. If $2500 is invested in this
account, what will the balance be after 15 years?
You try it!
1. 2 5 x − 6 = 8 x
2. 8 x + 5 = 16 2 x
3. 7 5 − x =
1
343
4. A money market account yields 1.8% annual interest compounded quarterly. If $4500 is invested, what will
the balance be after 5 years?
Assignment #2: page 488 #1-4, 6, 9-14, 20-22, 33-35
Honors Advanced Algebra
S8.3 Logarithms and Logarithmic Functions
Notes
Name: ________________________
Date: _________________________
Period: ________
WarmUp:
Solve.
1. 4 3x = 8 x + 5
2. 6 2 x =
Target Goal:
Evaluate logarithmic expressions.
1
216
In the equation x = b y , the variable y is called the logarithm of x . This is usually written as y = log b x , which
is read “ y equals log base b of x .”
LOGARITHM WITH BASE b
Let b and x be positive numbers, b ≠ 1 . The logarithm of x with base b is denoted log b x and is defined as
the exponent y that makes the equation b y = x true.
For x > 0 , there is a number y such that
if and only if
by = x
log b x = y
⇔ (exponential form)
(logarithmic form)
Example: If log 3 27 = y , then 3y = 27 .
It is often helpful to rewrite a logarithmic equation into exponential form or an exponential equation into
logarithmic form.
Write the equation in exponential form.
Ex 1. log 3 9 = 2
Ex 2. log10
1
= −2
100
Write the equation in logarithmic form.
1
Ex 3. 5 3 = 125
Ex 4. 27 3 = 3
Evaluate.
Ex 6. log 7
Ex 5. log 3 243
1
49
Ex 7.
log 1 27
3
Solve for x.
Ex 8. log x
1
= −3
64
Ex 9. log 4 ( x − 3) =
2
Ex 7.
log 2=
8 4x + 5
You try it!
1. Write log 1 125 = −3 in exponential form.
3
2
2. Write 4 = 8 in logarithmic form.
5
3. Evaluate log 8
1
.
16
Assignment #3: Worksheet #3
4. Solve. log 3 (2 x − 5) =
3
Honors Advanced Algebra
S8.4 Solving Logarithmic Equations
Notes
Name: ________________________
Date: _________________________
Period: ________
WarmUp:
1. Rewrite in exponential form.
3. Evaluate.
log 25 5 =
Target Goal:
2. Rewrite in logarithmic form.
1
2
4 −2 =
1
16
log 3 81
Solve logarithmic equations.
A logarithmic equation contains one or more logarithms. One way to solve a logarithmic equation involves
rewriting the equation into exponential form.
Solve the logarithmic equation.
Ex 1. log 8 x =
4
3
Ex 2. log16 x =
5
2
Ex 3. log x 8 =
3
2
Ex 4. log 4
1
=x
2
PROPERTY OF EQUALITY FOR LOGARITHMIC FUNCTIONS
Let b > 0 and b ≠ 1 . log b x = log b y if and only if x = y .
Example: If log 5 x = log 5 8 , then x = 8 . If x = 8 , then log 5 x = log 5 8 .
The Equality Property can be used to solve logarithmic equations if the logarithmic expressions have the same
(common) base.
Solve the logarithmic equation using the Equality Property.
Ex 5. log 9 2x = log 9 (24 − x )
(
Ex 6. log 5 x 2 = log 5 (x + 6 )
)
Ex 7. log 2 x 2 + 6x = log 2 (x − 4 )
You try it!
1. log 5 x = −2
2. log 3 2a = −2
4. log 9 (5x − 1) = log 9 (3x + 7 )
Assignment #4: page 504 #1-3, 8-19
3. log 2 x 16 = −2
5. log 6 (x − 3) = log 6 (2x )
Honors Advanced Algebra
S8.5 Properties of Logarithms
Notes
Name: ________________________
Date: _________________________
Period: ________
WarmUp:
Solve.
1. log 5 x = −3
Target Goals:
2. log 7 (x + 2 ) = log 7 4x
Simplify and evaluate expressions using the properties of logarithms.
Solve logarithmic equations using the properties of logarithms.
Since logarithms are exponents, the properties of logarithms can be derived from the properties of exponents.
PRODUCT PROPERTY OF LOGARITHMS
The logarithm of a product is the sum of the logarithms of its factors.
For all positive numbers a , b , and n , where n ≠ 1 , log n ab = log n a + log n b
Example: log 2 (5 )(6 ) = log 2 5 + log 2 6
You can use the Product Property to solve some logarithmic equations.
Solve the logarithmic equation using the Product Property.
Ex 1. log 4 x = log 4 3 + log 4 5
Ex 2. log 5 (y − 4 ) + log 5 (y + 4 ) = log 5 84
Ex 3. log10 x + log10 (x + 9 ) = 1
QUOTIENT PROPERTY OF LOGARITHMS
The logarithm of a quotient is the difference of the logarithms of the numerator and denominator.
For all positive numbers a , b , and n , where n ≠ 1 , log n
Example: log 2
5
= log 2 5 − log 2 6
6
a
= log n a − log n b
b
You can use the Quotient Property to solve some logarithmic equations.
Solve the logarithmic equation using the Quotient Property.
Ex 4. log 8 x − log 8 3 = log 8 5
Ex 5. log 3 4 = log 3 6 − log 3 x
POWER PROPERTY OF LOGARITHMS
The logarithm of a power is the product of the logarithm and the exponent.
For any real number p , and positive numbers m and b , where b ≠ 1 , log b m p = p log b m
Example: log 2 6 5 = 5 log 2 6
You can use the Power Property to solve some logarithmic equations.
Solve the logarithmic equation using the Power Property.
Ex 6. 2 log 4 x = log 4 9
Ex 7. 4 log 7 x = log 7
1
81
You try it!
1. log10 7 + log10 (n − 2 ) = log10 6n
2. 3log 7 x = log 7 27
3. log10 (m + 3) − log10 m = log10 4
4. log 9 x =
Assignment #5: page 512 #8-11, 23-26, 37-49 odd
1
1
log 9 144 − log 9 8
3
2
Honors Advanced Algebra
S8.6 Common Logarithms
Notes
WarmUp:
Solve.
1. log12 (x − 2 ) + log12 (x + 3) = log12 6
Target Goals:
Name: ________________________
Date: _________________________
Period: ________
2. 3log 5 2 − log 5 x = log 5 4
Solve exponential equations using common logarithms.
Evaluate logarithmic expressions using the change of base formula.
Base 10 logarithms are called common logarithms. Common logarithms are usually written without the
subscript 10.
log x = log10 x, x > 0
If both sides of an exponential equation cannot easily be written as powers of the same (common) base, one can
solve the equation by taking the logarithm of each side.
Solve the exponential equation using logarithms.
Ex 1. 5 x = 62
Ex 2. 7 x = 20
Ex 3. 3x +1 = 2 3x
Ex 4. 3x − 4 = 5 x −1
CHANGE OF BASE FORMULA
For all positive numbers a , b , and n , where a ≠ 1 and b ≠ 1 , log a n =
Example: log 3 11 =
log b n
log b a
log10 11 log11
=
log10 3
log 3
The change of base formula makes it possible to evaluate a logarithmic expression of any base by rewriting the
expression into one that involves common logarithms.
Evaluate using the change of base formula.
Ex 5. log 4 23
Ex 6. log14
2
3
You try it!
1. 25 x = 50
2. 5 4 y+1 = 32 2 y
3. log15 5
4. log11 104
2
Assignment #6: Worksheet #6
Honors Advanced Algebra
S8.7 Base e and Natural Logarithms
Notes
WarmUp:
1. Solve 3x = 14
Target Goals:
Name: ________________________
Date: _________________________
Period: ________
2. Evaluate log 6 83
Evaluate expressions involving the natural base and natural logarithm.
Solve exponential equations using natural logarithms.
Like π and 2 , the number e is an irrational number. The value is 2.71828... It is referred to as the natural
base, e . An exponential function with base e is called a natural base exponential function.
NATURAL BASE FUNCTIONS
The function f (x ) = e x is used to model continuous exponential growth.
The function f (x ) = e− x is used to model continuous exponential decay.
The inverse of a natural base exponential function is called the natural logarithm. This logarithm can be
written as log e x , but is more often abbreviated as ln x .
You can write a natural base exponential equation for a natural logarithmic equation:
ln 4 = x
⇔
log e 4 = x
⇔
ex = 4
Write the natural logarithmic equation in exponential form.
Ex 1. ln x = 0.5381
Ex 2. ln 25 = x
Write the natural base exponential equation in logarithmic form.
Ex 3. e x = 23
Ex 4. e4 = x
Equations involving base e are easier to solve using natural logarithms rather than common logarithms.
Solve. Round to the nearest ten-thousandth (4 decimal places).
Ex 5. 3e−2 x + 4 = 10
Ex 6. 5e 3x − 2 = 6
Solve. Round to the nearest ten-thousandth (4 decimal places).
Ex 7. 2 ln 5x = 6
Ex 8. 4 ln 3x = 11
CONTINUOUSLY COMPOUNDED INTEREST
You can calculate continuously compounded interest using the natural base exponential function:
A = Pert
where A is the amount in the account after t years, P is the principal amount invested, and r is the
annual interest rate.
Solve the continuously compounded interest problem.
Ex 9. An investment account pays 4.2% annual interest compounded continuously.
a. If $2500 is invested in this account, what will the balance be after 15 years?
b. How long will it take for any investment to double in value?
Assignment #7: page 529 #1-4, 9-14, 21-27 odd, 35-39 odd, 47, 53, 55