Exponential and Logarithmic Functions
Exponential and Logarithmic Functions
Objectives
In this unit, students will use exponential and logarithmic functions to solve real-world problems. Students will:
- simplify and evaluate exponential and logarithmic expressions.
- write and solve exponential and logarithmic equations.
- represent a set of data with an exponential equation and graph.
- determine the domain and range of exponential and logarithmic functions.
- identify the characteristics of exponential functions.
- use multiple representations for exponential and logarithmic functions.
- describe the effect of changing parameters in an exponential function.
Essential Questions
- How can we determine if a real-world situation should be represented by a quadratic, polynomial, or exponential function?
- How do you explain the benefits of multiple methods of representing exponential functions (tables, graphs, equations, and contextual situations)?
Related Unit and Lesson Plans
Related Materials & Resources
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Formative Assessment
-
View
Multiple-Choice Items
1. Which is the most simplified form of
?
A
B
C
D
2. What is the simplified form of 3log 2 – log 4?
A
log 2
B
log (3/2)
C
log 4
D
log 32
3. Which is equivalent to log2 (5/6) ?
A. log2 5 + log2 6
B. log2 5 – log2 6
C. log2 5 ÷ log2 6
D. log2 5 × log2 6
4. What does 0.5 represent in
?
A. 50% decrease
B. 50% increase
C. initial value
D. y-intercept
5. What does the number 6 represent in the following equation: y = 6(1.04)x?
A
6 % decrease
B
6 % increase
C
initial value
D
x-intercept
6. After two years you had $137.81 in an account, and after 4 years you had $151.94. What is your approximate annual interest rate?
A
5%
B
10.2%
C
1.05%
D
1.102%
7. Which is the asymptote in the graph of y = log(x – 5) + 2 ?
A
y = 2
B
x = 5
C
y = −2
D
x = −5
8. Where is the asymptote in the graph of the equation y = 5 + 2 log(x – 3)?
A
x = −3
B
x = 3
C
y = 5
D
y = −5
9. When solving the logarithmic equation
, which represents the correct second step in the solution process?
10. Solve for x using the change of base formula: 5x – 1 = 100
A
3.86
B
2.87
C
.349
D
1.349
11. Water is boiled to 100°C for a cup of tea. Five minutes later the tea is 88°C. The room temperature is 25°C. Which value represents the asymptote?
A
100
B
5
C
88
D
25
12. Water is boiled to 100°C for a cup of tea. Five minutes later the tea is 88°C. The room temperature is 25°C. At what rate is the tea cooling?
A
84%
B
96.6%
C
3.4%
D
1.034%
Multiple-Choice Answer Key
1. B
2. A
3. B
4. A
5. C
6. A
7. B
8. B
9. C
10. A
11. D
12. C
Short-Answer Items:
13. Simplify
. Show at least two steps.
14. A colony of bacteria began with 500. After 5 days there were 235. At what rate are the bacteria dying?
15. Graph the following logarithmic function: y = 2 – 4log9 (x – 1) and label all important components. Provide the domain and range.
16. The black bear population is growing. A study shows that there were 34 bears after 4 years and 57 bears after 8 years. How many bears were there when the study began?
Short-Answer Key and Scoring Rubrics:
13. Simplify
Scoring Rubric for Item 13
Points
Description
3
· The student writes the correct simplification of 3x and shows at least two steps.
2
· The student writes the correct simplification of 3x and shows at least one step.
1
· The student writes the correct simplification of 3x, but does not show any steps.
0
· The student does not provide an answer or writes the incorrect simplification.
14. A colony of bacteria began with 500. After 5 days there were 235. At what rate are the bacteria dying? 14%
15. Graph the following logarithmic function: y = 2 – 4log9 (x – 1) and label all important components. Provide the domain and range.)
Scoring Rubric for Item 15
Points
Description
3
· The graph has all of the following:
o axes are labeled
o vertical asymptote at x = 1 and is labeled
o x-intercept at (3, 0) and is labeled
o domain (–2 ≤ x ≤ 4) and range(0 ≤ y) are stated
2
· A graph is made but some parts are incorrect or unlabeled.
1
· A graph is made but not labeled.
0
· The student’s response is incorrect, irrelevant, too brief to evaluate, or missing.
16. The black bear population is growing. A study shows that there were 34 bears after 4 years and 57 bears after 8 years. How many bears were there when the study began?
20 bears
Performance Assessment:
Write a mini-mystery that would require using exponential and logarithmic functions to solve. Set the mystery up and then show your work to solve it. Use tables, graphs, and equations, as well as full sentences in your explanations.
Performance Assessment Scoring Rubric:
Points
Description
4
· The student’s mystery requires using exponential and logarithmic functions to solve. The student’s work has the following five components:
o an equation that represents the situation
o a table of values
o a graph that represents the situation
o all the steps to solving the mystery
o explanatory sentences (describing the mystery and how to solve it)
3
· The student’s work has four of the five necessary components.
2
· The student’s work has three of the five necessary components.
OR
· The student provides all necessary information except for how to solve the mystery.
1
· The student’s work has two of the five necessary components.
OR
· The student explains the mystery and how to solve it but does not provide the necessary visuals: equation, table, graph.
0
· The student’s response is incorrect, irrelevant, too brief to evaluate, or missing.
DRAFT 11/04/2010
