Lesson Plan 8.3: The Number e
Class: Honor Algebra 2
Grade Level: Sophomores and Juniors
Unit: Exponential and Logarithmic Functions
Teacher: Ms. Delaney Ludwig
Objectives
Students will be able to:
 Infer the use of the number e as the base of exponential functions.
 Apply the natural base e in real-life situations.
Essential Question
 When would you need to use the number e?
 How can different occupations apply the number e to their everyday life?
Anticipatory Set
Review/Brainstorming: (15 minutes)
For today’s anticipatory set I will ask students to recall special numbers in mathematics,
such as 𝜋. I will ask them if they can recall what 𝜋 is equal to (3.14). Following that
discussion, I will then ask them if they can think of any other special numbers we use to
solve equations with. After a few minutes of students answering, I will explain that there
is a number e that is a similar concept to having 𝜋 in an equation.
Teaching: Activities (35 minutes)
Using the Natural Base e:
I will explain that the history of mathematics is marked by the discovery of special
numbers. Like 𝜋 and i, the number e is denoted by a letter. The number called the natural
base e, or the Euler number, after its discoverer, Leonhard Euler. I will then draw the
table below on the board and walk through each box by asking students how they would
solve the equation.
n
𝑛
1
(1 + )
𝑛
101
2.594
102
2.704
103
2.716
104
2.718
105
2.718
106
2.718
Once the table is completed and students understand how we got each answer, I will ask
the class if the values in the table appear to be approaching a fixed decimal number. If so,
I will also ask them what is the number rounded to three decimal places. Following the
student’s discussion, I will explain that the example above shows that as n gets larger and
larger, the expression
1 𝑛
(1 + )
𝑛
gets closer and closer to 2.71828…, which is the value of e. After this concept is
understood by all of the students, I will explain that the natural base e is irrational.
Therefore, it is defined as follows:
1 𝑛
As 𝑛 approaches + ∞, (1 + ) approaches 𝑒 ≈ 2.718281828459.
𝑛
I will also explain to students what the symbol ≈ means just in case they have not seen it
before. After I have explained this concept, I will ask if there are any questions this far or
if anything seems to be confusing. I will then use some example problems to show
students exactly how they can use this property and how to evaluate an equation with the
number e.
Example problems:
 Simplifying Natural Base Expressions
a) 𝑒 3 ∗ 𝑒 4
Answer: To solve this equation, students would need to recall the properties of
multiplying numbers with exponents. Because both base values are the same, I
would explain that students need to simply add the exponents together like so:
𝑒 3 ∗ 𝑒 4 = 𝑒 3+4 = 𝑒 7
b)
(10𝑒 3 )
5𝑒 2
Answer: To solve this equation, students would again need to recall the
properties of exponents. I would then ask students how they might go about
solving this equation. Depending on their answers, I would then proceed to
solve the problem as follows:
10𝑒 3
= 2𝑒 3−2 = 2𝑒
5𝑒 2
c) (3𝑒 −4𝑥 )2
Answer: To solve this equation, students would again need to recall the
properties of exponents. I would then ask students how they might go about
solving this equation. Depending on their answers, I would then proceed to
solve the problem as follows:
(3𝑒 −4𝑥 )2 = 32 𝑒 −4𝑥∗2 = 9𝑒 −8𝑥 = 9/𝑒 8𝑥
I will ask if there are any questions this far or if anything seems to be confusing. If so, I
will go over the example problems again or I will change details to better help students
understand how to solve these problems.
Evaluating Natural Base Expressions:
I will then explain to students how to solve equations that use the number e by using a
calculator. I will use the smart board for this section, so students can see the actual
calculator on the screen. By using this type of technology, they will be able to see clearly
exactly what bottoms I am pressing on the calculator and why. I will first show them
where the specific buttons for these types of equations are located. Then, I will give them
example problems to try with their own calculators.
Example Problems:
 𝑒2
Answer: Students will need to press the keys (in this order) “2nd” , “e^x”, “2”, and
“enter”. Their answer should come out to 7.389056
 𝑒 −0.06
Answer: Students will need to press the keys (in this order) “2nd”, “e^x”, “(-)”, “.06”,
and “enter”. Their answer should come out to 0.941765.
Then, I will ask if there are any questions this far or if anything seems to be confusing. If
so, I will go over the example problems again or I will change details to better help
students understand how to solve these problems.
Exponential Growth or Exponential Decay:
A function of the form 𝑓(𝑥) = 𝑎𝑒 𝑟𝑥 is called a natural base exponential function. If a > 0
and r > 0, the function is an exponential growth function. And if a > 0 and r < 0, the
function is an exponential decay function.
Graphing Natural Base Functions:
 Graph the functions and state the domain and range.
a) 𝑦 = 2𝑒 0.75𝑥
Solution: I would draw the graph in the book explaining that because a=2 is
positive and r = .75 is positive, the function is an exponential growth function.
You can plot the points (0,2) and (1,4.23). I would also explain that these
coordinates were created by pluggining 0 and 1 for x and y values. The
domain is all real numbers, and the range is all positive numbers as well.
b) 𝑦 = 𝑒 −.5(𝑥−2) + 1
Solution: I would draw the graph in the book explaining that because a=1 is
positive and r = -.5 is negative, the function is an exponential decay function.
Translate the graph of 𝑦 = 𝑒 −.5𝑥 to the right 2 units and up 1 unit. The domain
is all real numbers and the range is y >1.
Using e in Real Life:
For this part of the class, I would tell students that these types of functions can come up
in real life situations. I would ask them if they could think of any situations this might
apply to. Based on the ideas students come up with, I would give them an example. My
example would be:
 I would first ask students to recall from Lesson 8.1 that the amount A in an
account earning interest compounded n times per year for t years is given by:
𝑟 𝑛𝑡
𝐴 = 𝑃 (1 + )
𝑛
Where P is the principal and r is the annual interest rate expressed as a decimal.
As n approaches positive infinity, the compound interest formula approximates
the following formula for continuously compounded interest:
𝐴 = 𝑃𝑒 𝑟𝑡
Another example I would give to show with actual numbers would be finding the balance
in an account.
 Say you deposit $1000 in an account that pays 8% annual interest compounded
continuously. What is the balance after 1 year?
Answer: To solve this equation, I would first explain that we need to use the
above formula we just created for continuously compounded interest. Then we
would note that P = 1000, r=.08, and t = 1. So, the balance at the end of 1 year is:
𝐴 = 𝑃𝑒 𝑟𝑡 = 1000𝑒 .08∗1 ≈ $1083.29
So in example 4 of Lesson 8.1, we found that the balance from daily
compounding is $1083.28. Therefore, continuous compounding earned only an
additional $.01.
I would then ask if there are any questions or certain areas the class would like me to go
over again. If not we will then move on to group work.
Closure
Group work: (25 minutes)
Once students seem to understand the material and I answered any questions there are, I
would then split students up into partners and have them work on problems from the
book to make sure they understand how to solve these problems. I would determine
partners by the time students went to bed the night before. Students will line up from the
time they went to bed from earliest to latest. I will then pair up students by who they are
standing next to. If there are an odd number of students, I will assign him or her to a
group.
Independent Practice
The problems from the book would consist of page 483-84, 17-39 odd, 49-55 odd, 67-75
odd. I would assign all of these problems because it would allow students to evaluate
expressions and construct answers. Students would also be able to check the back of the
book’s answer key to see if they are getting the correct answer by assigning the odd
numbers. Students would be working with a partner on these problems, this way they can
help each other out if struggling. I would be walking around the room at this time to
make sure students are doing problems correctly and answering any questions. If students
seem to be finishing early, I would start going over the answers by asking student what
answers they got for each problem. If students seem to be struggling with certain
problems from the book, I would do a problem at the board with the entire class as
another example on how to solve these types of equations. I would then inform the class
that if they did not get these problems finished in class, it is homework.
Assessment
As formative assessments, I would have students convert what they had learned in this
lesson into a text message to one of their friends. I would pass the form attached to this
lesson to each student and ask them to summarize the lesson in a simple text message. I
would encourage them to use the abbreviations of text messages and to keep it simple by
writing down the most important thing they learned. I would give them the last 10-15
minutes of class because they might need some extra time to think and write out their
ideas. If they seem to be finishing up early I would collect the sheets and remind them of
the homework.
Materials
Copies of exit sheet
Duration
This lesson will last approximately 90 minutes.
Modified from Madeline Hunters Lesson Plan Design
Lesson Plan 8.4: Logarithmic Functions
Class: Honor Algebra 2
Grade Level: Sophomores and Juniors
Unit: Exponential and Logarithmic Functions
Teacher: Ms. Delaney Ludwig
Objectives
Students will be able to:
 evaluate logarithmic functions
 illustrate real-life situations, such as evaluating the slope of a beach
Essential Question
 When would you need to use logarithmic functions?
 How can different occupations apply logarithmic functions to their everyday life?
Anticipatory Set (15 minutes)
For today’s anticipatory set I will ask students to recall the idea of exponential functions.
I will ask them to define the terms “exponential decay” and “exponential growth”. After
students have defined these terms, I will explain that logarithmic functions are the
inverses of exponential functions. Logarithms are found in applications such as
compound interest, earthquake magnitudes, and the pH of solutions in chemistry. I would
explain that we know 22 = 4 and 23 = 8. However, for what value of x does 2𝑥 =
6? Because 22 < 6 < 23 , you would expect x to be between 2 and 3. To find the exact xvalue, mathematicians defined logarithms. In terms of a logarithm, 𝑥 = log 2 6 ≈ 2.585.
Teaching: Activities (35 minutes)
Rewriting Logarithmic Equations:
Once students seem to be recognizing the similarities between exponential functions and
logarithmic functions, I will begin to define the definition of logarithm with base b. I
would begin by saying:

Let b and y be positive numbers, but b cannot equal 1. The logarithm of y with
base b is written by log 𝑏 𝑦 and is defined as log 𝑏 𝑦 = 𝑥 if and only if 𝑏 𝑥 = 𝑦. The
expression log 𝑏 𝑦 is read as “log base b of y.”
I will then conclude that this definition tells you that the equations log 𝑏 𝑦 = 𝑥 𝑎𝑛𝑑 𝑏 𝑥 =
𝑦 are equivalent. The first is in logarithmic form and the second is in exponential form.
Given an equation in one of these forms, you can always rewrite it in the other form.
Example Problems:
 Write the logarithmic equation in exponential form:
a) log 2 32 = 5
Answer: 25 = 32
b) log 5 1 = 0
Answer: 50 = 1
c) log10 10 = 1
Answer: 101 = 10
d) log10 . 01 = −1
Answer: 10−1 = 0.1
e) log1/2 2 = −1
1 −1
Answer: (2) = 2
I will then ask if there are any questions or confusion. If so I will explain the material
again or write another example problem.
Special Logarithmic Values:
Next, I will identify to students that examples b) and c) are actually two special
logarithmic values that they should memorize. I will clarify by saying:
 Let b be a positive real number such that b does not equal 1.
o Logarithm of 1: log 𝑏 1 = 0 because 𝑏 0 = 1
o Logarithm of base b: log 𝑏 𝑏 = 1 because 𝑏1 = 𝑏
I will then again ask if there are any questions or confusion. If so I will try to reiterate
these definitions in a simpler way. If not I will move on to other example problems.
Evaluating Logarithmic Expressions:
 Evaluate the expression:
a) log 3 81
Answer: think 3 to what power gives you 81?
34 = 81, 𝑠𝑜 log 3 81 = 4
b) log 5 0.04
Answer: think 5 to what power gives you 0.04?
5−2 = 0.04, 𝑠𝑜 log 5 0.04 = −2
c) log1/2 8
1
Answer: think 2 to what power gives you 8?
1−3
= 8, 𝑠𝑜 log1/2 8 = −3
2
d) log 9 3
Answer: think 9 to what power gives 3?
1
1
92 = 3, 𝑠𝑜 log 9 3 =
2
I will explain that to help students find the value of log 𝑏 𝑦, ask yourself what the power
of b gives you y. I will then ask if there are any questions or confusion. If students seem
to be having trouble with evaluating the expression I will have other example problems to
work with.
Evaluating Common and Natural Logarithms:
When it seems that students are able to define logarithmic functions and exponential
functions, I will then introduce specific terms for them to use while constructing these
concepts. I will explain that the logarithm with base 10 is called the common logarithm.
It is written by log10 or simply by log. The logarithm with base e is called natural log. It
can be written by log 𝑒 but it is more often known as ln.
 Common logarithm: log10 𝑥 = log 𝑥
 Natural logarithm: log 𝑒 𝑥 = ln 𝑥
After writing these concepts on the board I will give students two example problems. I
will then explain to students how to solve equations that have logarithmic functions by
using a calculator. I will use the smart board for this section, so students can see the
actual calculator on the screen. By using this type of technology, they will be able to see
clearly what buttons I am pressing on the calculator and why. I will first show them
where the specific buttons for these types of equations are located. Then, we will do the
example problems together in class.
Example Problems:
a) log 5  press keys “LOG”, “5”, and “enter”.
Answer: 0.698970
b) ln 0.1 press keys “LN”, “.”, “1”, “enter”.
Answer: -2.302585
At this point, I will ask students if there are any questions over anything we have talked
about this far. If so I will go over those topics again or give another example problem. If
not, I will move on to evaluating a logarithmic function in a real life situation.
Evaluating a Logarithmic Function/Problem Solving:
For this part of the class, I would tell students that these types of functions can come up
in real life situations. I would ask them if they could think of any situations this might
apply to. Based on the ideas students come up with, I would give them an example. My
example would be:
 The slope s of a beach is related to the average diameter d (in millimeters) of the
sand particles on the beach by this equation:
𝑠 = 0.159 + 0.118 log 𝑑
Find the slope of a beach if the average diameter of the sand particles is 0.25
millimeter.
Answer: If d= 0.25, then the slope of the beach is:
𝑠 = 0.159 + 0.118 log 0.25 (substitute 0.25 for d.)
𝑠 = 0.159 + 0.118(−0.602) (use a calculator to solve)
𝑠 ≈ 0.09
(simplify)
The slope of the beach is about 0.09. This is a gentle slope that indicates a rise of
only 9 meters for a run of 100 meters.
Closure (25 minutes)
Once students seem to be comfortable with logarithmic functions, I will break them off
into partners. Their partners will be determined by their initials. I will have students find
another classmate with the same letter as their first name. For example, since my name is
Delaney, I would need to find a partner with the letter D. If a student cannot find
someone with the same letter as their first name, I will tell them to try matching last name
letters or middle name letters. If students still are not paired up. I will assign each student
a person. If there are an odd number of students then there will be one group of 3. Once
students have found a partner or group, I will pass out a worksheet. They will work on
the worksheet together. By working together, they will be able to compare each other’s
ideas and answers. Students will also be able to explain and extend the point of the lesson
to their partner if they do not understand the material or are having a tough time solving a
problem.
Independent Practice
Students would have the rest of the class time to work on the worksheet. Every student
would need to turn in their own worksheet, but they would be encouraged to work
together or ask each other for help if needed. I would be walking around checking on
students. If students seemed confused on specific problems, I would ask the class to pick
a problem they would like me to do on the board. By analyzing the problem at the board,
this would help clarify some confusion and hopefully it would set students on the right
track to finishing the worksheet successfully. I would then inform the class that if they
did not get this worksheet finished in class, it is homework and would be collected next
class period.
Assessment
As formative assessments, I would give each student a ticket form and ask them to briefly
write down what the most important thing they learned today was. I would ask them to
write one question they still have about anything we covered during this class period. I
would give them the last 10-15 minutes of class because they might need some extra time
to think and write out their ideas. If they seem to be finishing up early I would collect the
forms and remind them of the homework.
Materials
Worksheet
Exit forms
Duration
This lesson will last approximately 90 minutes.
Modified from Madeline Hunters Lesson Plan Design