Algebra 2 Unit 6 - Exponential & Logarithmic Functions Packet - Dam

UNIT 6:
Exponential & Logarithmic

Functions

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SUCCESS CRITERIA

I can… sketch exponential and logarithmic functions.

I can explain… the inverse relationships between exponential and logarithmic functions.

I can justify… my steps when solving exponential and logarithmic equations.

I can apply… exponential and logarithmic concepts to analyze a real-life scenario.

LESSON TABLE OF CONTENTS PG #
6.1 3-5
6.2 TITLE 6
6.3 Exponential Functions 7-8
6.4 Natural Base Exponential Functions 9
6.5 Compound Interest 10
6.6 Logarithmic Functions 11
6.7 Transformations of Exponential & Logarithmic Functions 12
6.8 Logarithmic Properties 13
Exponential & Logarithmic Equations 14-16
Modeling with Exponential & Logarithmic Functions
Unit 6 Review 2

Use Desmos to match each exponential function to its correct graph.

Identify the key features of the exponential functions above. D E F

ABC 3
Domain

Range

y-int

What do you notice about:
● The domain of exponential functions?

● The range of exponential functions?

● The increasing/decreasing behavior of exponential functions?

Lesson 6.1 - Exponential Functions

Exponential Functions

1) Yahir plans to purchase a car that depreciates (loses value) at a rate of 12% per year. The initial
cost of the car is $19,000 dollars. How much will the car be worth after 3 years?

2) Diana is opening a savings account at her local bank. She invests an initial amount of $5000. Her
bank gives her 5 ¼% interest on her account each year. How much money will Diana have after
25 years? How much interest has Diana accumulated over that time?

Practice

1) The number of bacteria (in thousands) in a culture can be approximated by the model =
100(1.99) , where is the number of hours the culture is incubated.
a) Tell whether the model represents exponential growth or decay.
b) Identify the hourly percent increase or decrease in the number of bacteria.
c) Estimate when the number of bacteria will be 1,000,000.

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2) The value of a rare coin (in dollars) can be approximated by the model = 0.25(1.06) , where
is the number of years since the coin was minted.
a) Tell whether the model represents exponential growth or decay.
b) Identify the annual percent increase or decrease in the value of the coin.
c) What was the original value of the coin?
d) Estimate when the value of the coin will be $0.60.

3) The value of a truck (in dollars) can be approximated by the model = 54,000(0.80) , where
is the number of years since the truck was new.
a) Tell whether the model represents exponential growth or decay.
b) Identify the annual percent increase or decrease in the value of the truck.
c) What was the original value of the truck?
d) Estimate when the value of the truck will be $30,000.

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Lesson 6.2 - Natural Base Exponential Functions
Natural Base Exponential Functions

Identify whether the function is an exponential growth or decay and its percent of change.
1) = 2−3

2) = 1500(1 + 0.032 )4

4

3) = 1200 0.218

4) = 1200 (0.73 )

−

5) = 1800(0.74) 8

6) = 700( 1

2 ) 61

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Lesson 6.3 - Compound Interest Compounded Continuously
Compounded n Times Per Year

1) Grace puts $20,000 in a savings account paying 8% interest compounded monthly. At this rate,
how much money will she have in the account after 40 years?

2) Jeremy wants to have $20,000 for retirement in 45 years. He invests in a mutual fund paying an
average of 9.5% each year compounded quarterly. How much should he deposit into his mutual
fund?

3) Sarah wishes to turn her $10,000 investment into $100,000 in 20 years. How much interest does
she need to receive annually to reach her goal?

4) Megan invests $50,000 into an index annuity that’s averaging 8.4% per year compounded
semiannually. At this rate, how many years will it take her account to reach $1,000,000?

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5) Ellen invests $100,000 in an account paying 7.5% interest compounded continuously. How much
money will be in her account in 30 years?

6) Anthony wants to have $1,500,000 in 50 years. How much should he invest now in an account
paying 12% interest compounded continuously?

7) Nathan invests $5000 in an account paying 11% interest compounded continuously. How long will
it take him to have $2,000,000?

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Lesson 6.4 - Logarithmic Functions

Logarithmic Functions

Rewrite each equation in logarithmic form. Rewrite each equation in exponential form.
1) 3 = 10 4) 6 =

2) 2 −5 = 50 5) 5 = 4

3) 12 = 144 6) 3( − 2) = 4

Evaluate each logarithm.

7) 2 1 8) 3 = 4
16 9) 27 = 3

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Lesson 6.5 - Transformations of Exponential & Logarithmic Functions

Transformation Function Notation Example ( ) = 4 ( ) =

Horizontal 3 units right
Translation 3 units left

Vertical 3 units up
Translation 3 units down

Reflection over y-axis
over x-axis

Horizontal H compress by factor of 3
Dilation H stretch by factor of 3
Vertical
Dilation V compress by factor of 3
V stretch by factor of 3

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Describe and graph each transformation.

1) ( ) = 6 2) ( ) = 3) ( ) =
( ) = 6 + 6 ( ) = −4 ( ) = −3 ( − 2)

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Lesson 6.6 - Logarithmic Properties

Product Property
Quotient Property
Power Property
Change of Base Formula

Expand each logarithmic expression. Condense each logarithmic expression.
1) (5 ) 7) 3 − 2

2) ( + 1)( − 2) 8) 73 − 5
9) 3 + 9
3) 9



4) 3 10) 69 + 2 6 1 − 3 6
+1 3

5) 2 Evaluate each logarithm.
11) 53
6) 6 −2
12) 211

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Lesson 6.7 - Exponential & Logarithmic Equations 2) 2 = 17

Solve each equation.
1) 2 = 16

3) 32 −7 = 27 4) 3 +2 = 2 −1

5) 2( − 4) = 3 6) 2 + 2( − 3) = 2

7) = 63 8) 10 +3 + 5 = 105
9) (3 − 2) = 1 10) ( + 1)2 = 2

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Lesson 6.8 - Modeling with Exponential & Logarithmic Functions

Write an exponential function whose graph passes through the given points.
1) (1,12) (3,108)

2) (−1,2) (3,32)

3) (2,9) (4,324)

4) The table below represents the total number of goggles sold at a given number of months.

Months 1 2 3 4 5 6

Goggles sold 28 47 64 79 97 107

5) The table below represents an exponential relationship between x and y.

x 0123
y 8 12 18 27

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Unit 6 Review

1) Determine whether each table contains data that can be modeled by an exponential function.
For those that are exponential, write an equation to show the relationship between x and y.

a)
x0123

y 3 6 12 24

b)
x0123

y2468

c)
x0123

y 108 36 12 4

2) Determine whether each function is increasing or decreasing. Then identify the domain, range,
and y-intercept of the function.

a) = 4(2)

3

b) = −3 ⋅ 4

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3) Let ( ) = 2 ⋅ 4 +3 − 5

a) Describe ( ) as a transformation of ( ) =
4 .

b) Graph ( ) using transformations.

c) What is the horizontal asymptote of ( )?

4) Rewrite each exponential equation 5) Rewrite each logarithmic equation
as a logarithmic equation. as an exponential equation.
a) 10 = 1000 a) 100 = 2

b) 10− 4 = 1 b) 100,000 = 5

10,000 c) 1 = −5

c) 107 = 10,000,000 10,000

6) Evaluate each logarithmic expression without using a calculator.
a) 1000

b) 1

c) 2 + 50

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7) Let ( ) = 2( − 1) + 3
a) Sketch the parent graph of ( ) and
the graph of ( ).

b) What is the vertical asymptote of ( )?

8) Alyssa deposits $10,000 in a savings account that pays 8.5% interest per year, compounded
quarterly. She does not deposit more money and does not withdraw any money.
a) Write an equation to represent the amount in Alyssa’s account after 3 years.
b) Find the total amount Alyssa will have in her account after 3 years.

9) How long would it take an investment of $6500 to earn $1200 interest
if it is invested in a savings account that pays 4% annual interest
compounded quarterly? Show the solution both algebraically and
graphically.

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