Logarithms Functions
Author: Hannah Knisely, Kent Island High School, Queen Anne’s County Public Schools
Background Information
Subject:
Algebra II
Grade Band:
9-12
Duration:
90 Minutes
Overview:
Students will discover patterns within logarithmic functions, thus discovering logarithmic
properties. They will analyze and apply logarithmic functions to sound frequencies and
earthquakes. A STEM Specialist will help students interpret their findings with the properties
of logarithms as well as engage students in hands-on learning experiences that allow them to
draw connections between course content ant the work performed by STEM professionals.
Teachers should familiarize themselves with the Properties of Logarithms:
 𝑙𝑜𝑔𝐴𝐵 = 𝑙𝑜𝑔𝐴 + 𝑙𝑜𝑔𝐵
𝐴
 𝑙𝑜𝑔 𝐵 = 𝑙𝑜𝑔𝐴 − 𝑙𝑜𝑔𝐵
 𝑛𝑙𝑜𝑔𝐴 = 𝑙𝑜𝑔𝐴𝑛
 𝑙𝑜𝑔𝑏 1 = 0
 𝑙𝑜𝑔𝑏 𝑏 = 1
 𝑙𝑜𝑔𝑏 𝑏 𝑥 = 𝑥
 𝑏𝑙𝑜𝑔𝑏 𝑥 = 𝑥
1
 𝑙𝑜𝑔𝑥 ( 𝑎) = −𝑎
𝑥
Background Information:
Additionally, teachers should understand the scales for both decibels and magnitude and how
they relate to logarithms.
dB
1
10
20
30
40
50
60
70
80
90
100
Sound
100 101
102
103
104
105
106
107
108
109
1010
Intensity
The Richter scale was developed to assign a single number to quantify the energy released
during an earthquake. The scale is a base-10 logarithmic scale. The magnitude is defined as
the logarithm of the ratio of the amplitude of waves measured by a seismograph to an
arbitrary small amplitude. An earthquake that measures 5.0 on the Richter scale has a
Logarithms Functions
Background Information
shaking amplitude 10 times larger than one that measures 4.0, and corresponds to a 31.6
times larger release of energy.
STEM Specialist Connection:
Enduring Understanding:
Essential Questions:
Student Outcomes:
Product, Process, Action, Performance,
etc.:
The STEM Specialist can:
 Help students decipher patterns between logarithmic expressions.
 Engage students in hands-on learning experiences on the application of logarithms in the
workplace.
STEM Specialists can be found at www.theSTEMnet.com
 Logarithms allow us to solve for exponential equations in which the variable is in the
exponent.
 Logarithmic functions can be used to model seismic activity, sound intensity,
magnitude of earthquakes, compute compound interest in terms of time, calculate
population growth, radioactive decay, etc.
 Logarithms were developed to multiply large numbers without the use of a calculator.
1. How can logarithms be used to solve exponential functions in which the variable is
found in the exponent?
2. How do STEM professionals apply logarithmic functions to develop solutions to
problems?
Students will be able to:
1. Convert exponential functions to logarithmic functions and vice versa.
2. Solve base 10 logarithms.
3. Apply the properties of logarithms to simplify and expand different logarithmic
functions.
4. Interpret and solve real world situations in which logarithms would be necessary to
solve the problem.
Audience:
☒Peers
☒Experts /
Practitioners
Students will work independently to derive the properties of logarithmic
functions and apply them to real-world problems.
☒Teacher(s)
☐School
Community
☐Other______
Logarithms Functions
Standards Addressed in the Unit:
Background Information
Domain: Modeling with Functions
Cluster: Construct and compare linear, quadratic, and exponential models and solve
problems
Standard: F.LE.A.4 For exponential models, express as a logarithm the
solution to abct=d where a, c, and d are numbers and the base b is 2, 10, or e;
evaluate the logarithm using technology.
Cluster: Build a function that models a relationship between two quantities.
Standard: F.BF.A.1: Write a function that describes a relationship between
two quantities.
Equipment:
 computer with internet access
 projector
 Scientific or Graphing calculator
 Computers for each student (optional)
 Earphones (optional)
Website*
http://www.dangerousdecibels.org/virtualexhibit/
Suggested Materials and Resources:
* Students are linked to an online site for the engagement activity. The site has been chosen for its content and
grade-level appropriateness. Teachers should preview website before introducing the activities to students and
adhere to their school system’s policy for internet use.
People, Facilities:
 STEM Specialist
 Computer lab (optional)
Materials (rubrics, worksheets, PowerPoints, answer keys, etc.):
 Logarithms PowerPoint
 Discovering Properties of Logarithms
 Homework
 Homework (Answers)
 Calculator
 Computer (with internet access)
 Projector
Logarithms Functions
Learning Experience
5E Component
Identify the 5E
component
addressed for the
learning
experience. The 5E
model is not linear.
☒Engagement
☐Exploration
☐Explanation
☐Extension
☐Evaluation
Details
Standards for Mathematical
Practice
Materials:
 Computer with internet access
 Projector
 Logarithms PowerPoint
 Individual Computers for students (optional)
 Earphones (optional)
☐Make sense of problems
and persevere in solving
them.
Preparation:
 Explore http://www.dangerousdecibels.org/virtualexhibit/ on your own.
Understand how the website works and make a decision whether you want
your students to explore the site individually or as a class.
 Familiarize yourself with decibels and sound intensity. Be able to explain
the different scales to your students.
 Use PowerPoint slides 1-4 as a guide through the engagement portion of
the lesson.
☐Construct viable arguments
and critique the reasoning
of others.
Facilitation of Learning Experience:
 Begin your lesson with the Dangerous Decibels Activity
(http://www.dangerousdecibels.org/virtualexhibit/). Have students—
individually or as a class—explore the website. You can choose to do as
many (or as little) activities as your class time allows. Be sure not to leave
out the “measuring sound” section as that is the piece that explains the
decibel scale. Students should be carefully monitored during this activity
since the sounds can be pretty loud.
 If you choose to allow the students to explore the website on their own,
you will need to monitor their use of time in each section.
 As a class, discuss the results from each section of the Dangerous
☐Attend to precision.
☐Reason abstractly and
quantitatively.
☐Model with mathematics.
☒Use appropriate tools
strategically.
☐Look for and make use of
structure.
☐Look for and express
regularity in repeated
reasoning.
Logarithms Functions
Learning Experience
5E Component
Identify the 5E
component
addressed for the
learning
experience. The 5E
model is not linear.
Details

☐Engagement
☐Exploration
☒Explanation
☐Extension
☐Evaluation
Standards for Mathematical
Practice
Decibels activity. If you are completing the entire activity as a class this
should be done throughout. If the students are doing this activity
individually, you will need to discuss each element of their exploration.
Compare and contrast how each student did with the activity.
Continue your discussion on decibel levels by showing the next 3 graphics
on the PowerPoint—infographic (bubbles), lesser water boatman graph of
noise levels, and the frequencies heard by different animals.
Materials:
 Computer
 Projector
 Logarithms PowerPoint
Preparation:
Use PowerPoint slides 5-11 as a guide through the explanation section.
Facilitation of Learning Experience:
 Discuss the origin of logarithms—where they come from, what they were
(still are) used for, etc.
 Be sure to indicate how you switch from logarithmic to exponential form
and vice versa. The equation for this process is found at the bottom of
slide 8. Continue this discussion by working through a few examples with
your students (found on the next two slides).
 Next, discuss the common bases used in everyday life (Base 10, 2, and e)
 Ask the students when they have used logarithms (such as in chemistry
class) and why logarithms were used.
☐Make sense of problems
and persevere in solving
them.
☐Reason abstractly and
quantitatively.
☐Construct viable arguments
and critique the reasoning
of others.
☐Model with mathematics.
☐Use appropriate tools
strategically.
☐Attend to precision.
☒Look for and make use of
Logarithms Functions
Learning Experience
5E Component
Identify the 5E
component
addressed for the
learning
experience. The 5E
model is not linear.
Details
Standards for Mathematical
Practice
structure.
☒ Look for and express
regularity in repeated
reasoning.
☐Engagement
☒Exploration
☐Explanation
☐Extension
☐Evaluation
Materials:
 Discovering Properties of Logarithms worksheet
 Pencil/Pen
 Calculator
 Computer
 Projector
 Logarithms PowerPoint
Preparation:
 Use PowerPoint slides 12-17 as a guide through the exploration section.
 Complete the Discovering Properties of Logarithms worksheet in advance
to familiarize yourself with the patterns the students should discover.
Facilitation of Learning Experience:
 Provide each student with a copy of discovering properties of logarithms
worksheet. Allow students to work in partners to complete the assignment.
Students should notice the patterns that develop in the first column
between addition and multiplication. They should also notice the pattern
between subtraction and division in the second column. Finally, they
should notice that exponents are directly related to multiplying by a
coefficient of the same numerical value.
 Discuss as a class the discoveries that were made among the pairs in the
class.
 Continue that discussion with slides 12 - 15 of the PowerPoint. The
☒Make sense of problems
and persevere in solving
them.
☒Reason abstractly and
quantitatively.
☒Construct viable arguments
and critique the reasoning
of others.
☐Model with mathematics.
☒Use appropriate tools
strategically.
☒Attend to precision.
☒Look for and make use of
structure.
☐ Look for and express
regularity in repeated
Logarithms Functions
Learning Experience
5E Component
Identify the 5E
component
addressed for the
learning
experience. The 5E
model is not linear.
Details

☐Engagement
☐Exploration
☐Explanation
☒Extension
☐Evaluation
properties the students should have discovered are present here. There
are a few other properties that should be noted in the student’s notes that
were not discovered with the activity. Be sure they are included in their
notes as shown on slide 15.
Use these properties to work through the next two examples as a class.
These examples are worked through on slides 16 & 17.
Materials:
 Logarithms PowerPoint
 Computer
 Projector
Preparation:
Use slides 18 – 23 as a guide through the extension section of the lesson.
Facilitation of Learning Experience:
 As a class, work through the next two examples (slides 19-23) together.
The first example revisits the idea of decibels and asks the students to
determine just how much louder one decibel level is than another in terms
of relative intensity.
 Review the earthquake graph with your students. Point out different
magnitude levels and show them the comparable energy equivalent. Many
will be surprised by the amount of energy emitted from an earthquake and
the comparable events.
 The earthquake example compares two different earthquakes magnitudes
and asks the students to solve just how much more energy one
earthquake had than the other.
Standards for Mathematical
Practice
reasoning.
☒Make sense of problems
and persevere in solving
them.
☒Reason abstractly and
quantitatively.
☒Construct viable arguments
and critique the reasoning
of others.
☐Model with mathematics.
☒Use appropriate tools
strategically.
☒Attend to precision.
☐Look for and make use of
structure.
Logarithms Functions
Learning Experience
5E Component
Identify the 5E
component
addressed for the
learning
experience. The 5E
model is not linear.
Details
Standards for Mathematical
Practice
☒ Look for and express
regularity in repeated
reasoning.
☐Engagement
☐Exploration
☐Explanation
☐Extension
☒Evaluation
Materials:
 Logarithmic Functions Homework
 Logarithmic Functions Answer Key
 Pencil/Pen
 Calculator
Preparation:
☒Make sense of problems
and persevere in solving
them.
☒Reason abstractly and
quantitatively.
Contact the STEM Specialist in advance to co-plan the learning experience
for students. STEM Specialist can be found at wwww.theSTEMnet.com. A
description of the ability level of the students, as well as some of the prior
knowledge your students may have of logarithmic functions may be helpful
to the STEM Specialist prior to the visit.
☐Construct viable arguments
and critique the reasoning
of others.
Facilitation of Learning Experience:
☒Use appropriate tools
strategically.


Have the STEM Specialist engage students in the predetermined learning
experience emphasizing how logarithmic functions are used in the
workplace.
After the STEM Specialist has engaged students in the learning
experience, provide each student a copy of the homework assignment to
work on independently after class. At the start of the next class, you can
review answers or collect papers for grading.
☒Model with mathematics.
☒Attend to precision.
☐Look for and make use of
structure.
☒ Look for and express
regularity in repeated
reasoning.
Logarithms Functions
Supporting Information
Struggling Learners
Students should be provided with copies of notes or a note guide to follow along with the
PowerPoint. Additionally, students could work with the teacher or STEM specialist in the
discovery activity to better guide them to the patterns present. A list of the different
properties of logarithms can (should) be provide to struggling learners to help aide them
in solving difficult equations.
English Language Learners
Interventions/Enrichments
Identify interventions and enrichments for
diverse learners.
Similarly to struggling learners accommodations should be provided. A vocabulary list
may be helpful to ELL students (logarithm, exponent, magnitude, decibel, etc.) Many
words in this lesson are comparable to European languages because of the origin of the
words therefore, vocabulary may be helpful but is not necessarily needed in all ELL
students.
Gifted and Talented
Expand student knowledge by providing more complex, challenging questions. Begin
with the basics that all students are learning. Give them more advanced questions as
practice problems if they should finish early. Match these students with some of the
struggling learners for the discovery lesson to aide those students as well as enhance
the understanding of your gifted and talented students. As an additional extension, have
the gifted and talented students research magnitudes of different earthquakes around
the world and write their own example problems (providing the answers and work along
with it).
Discovering Properties of Logarithms
Investigation Part One
Investigation Part Two
Investigation Part Three
Use your calculator to find the
Use your calculator to find the
Use your calculator to find the
following common logs:
following common logs:
following common logs:
 log3 + log 4
 log6 - log2
 3log2
 log 4 + log5
 log20 - log5
 2 log5
 log5 + log6
 log12 - log6
 3log 4
 log3 + log5
 log3 - log 4
 0.5log16
 log3 + log6
 log9 - log27
 -2 log3
 log 4 + log6
 log 4
 log64
 log20
 log(0.75)
 log25
 log15
 log2
 log 4
 log24
 log3
æ 1ö
 log ç ÷
è 9ø
 log18
æ 1ö
 log ç ÷
è 3ø
 log8
 log12
 log30
Write down any patterns or connections you recognize between your answers. What relationships can you identify?
Develop a general rule to write
log A + log B as a single logarithm, for
all positive values of A and B.
Develop a general rule to write
log A - log B as a single logarithm, for
all positive values of A and B.
Test your general rule on several other values for A and B.
Develop a general rule to write
n log A as the logarithm of a single
constant, for all positive values of A
and all real values of n.
Test your general rule on several other
values for A and n.
Logarithmic Functions Homework
Write the following equations in logarithmic/exponential form
1. 53 = 125
2. 𝑙𝑜𝑔10 0.00001 = −5
1
3. 77765 = 6
3
4. 𝑙𝑜𝑔32 8 = 5
Solve each equation.
5. 𝑙𝑜𝑔4 𝑥 = 2
11. 𝑙𝑜𝑔2 (𝑥 + 4) − 𝑙𝑜𝑔2 (𝑥 − 3) = 3
6. 𝑙𝑜𝑔1 27 = 𝑥
12. 𝑙𝑜𝑔5 10 + 𝑙𝑜𝑔5 12 = 3𝑙𝑜𝑔5 2 + 𝑙𝑜𝑔5 𝑎
3
3
7. 𝑙𝑜𝑔𝑏 1024 = 5
8. 𝑙𝑜𝑔7 (𝑥 + 6) = 𝑙𝑜𝑔7 (8𝑥 + 20)
9. 𝑙𝑜𝑔4 5 + 𝑙𝑜𝑔4 𝑥 = 𝑙𝑜𝑔4 60
10. 𝑙𝑜𝑔2 𝑞 − 𝑙𝑜𝑔2 3 = 𝑙𝑜𝑔2 7
13. An equation for loudness, in decibels, is 𝐿 = 10𝑙𝑜𝑔10 𝑅, where R is the relative
intensity of the sound. Sounds that reach levels of 120 decibels or more are
painful to humans. What is the relative intensity of 120 decibels?
14. An earthquake rated at 3.5 on the Richter scale is felt by many people, and an
earthquake rated at 4.5 may cause local damage. The Richter scale magnitude
reading 𝑚 is given by 𝑚 = 𝑙𝑜𝑔10 𝑥, where 𝑥 represents the amplitude of the
seismic wave causing ground motion. How many times greater is the amplitude
of an earthquake that measures 4.5 on the Richter scale than on the measures
3.5?
15. Rob is solving a problem involving logarithms. He does everything correctly
except for one thing. He mistakenly writes 𝑙𝑜𝑔2 𝑎 + 𝑙𝑜𝑔2 𝑏 = 𝑙𝑜𝑔2 (𝑎 + 𝑏).
However, after substituting the values for 𝑎 and 𝑏 in his problem, he amazingly
still gets the right answer! The value of 𝑎 was 11. What must the value of 𝑏 have
been?
Logarithmic Functions Homework
Write the following equations in logarithmic/exponential form
1. 53 = 125
2. 𝑙𝑜𝑔10 0.00001 = −5
1
3. 77765 = 6
4. 𝑙𝑜𝑔32 8 =
3
5
Solve each equation.
5. 𝑙𝑜𝑔4 𝑥 = 2
9. 𝑙𝑜𝑔4 5 + 𝑙𝑜𝑔4 𝑥 = 𝑙𝑜𝑔4 60
6. 𝑙𝑜𝑔1 27 = 𝑥
10. 𝑙𝑜𝑔2 𝑞 − 𝑙𝑜𝑔2 3 = 𝑙𝑜𝑔2 7
3
3
7. 𝑙𝑜𝑔𝑏 1024 = 5
8. 𝑙𝑜𝑔7 (𝑥 + 6) = 𝑙𝑜𝑔7 (8𝑥 + 20)
11. 𝑙𝑜𝑔2 (𝑥 + 4) − 𝑙𝑜𝑔2 (𝑥 − 3) = 3
12. 𝑙𝑜𝑔5 10 + 𝑙𝑜𝑔5 12 = 3𝑙𝑜𝑔5 2 + 𝑙𝑜𝑔5 𝑎
13. An equation for loudness, in decibels, is 𝐿 = 10𝑙𝑜𝑔10 𝑅, where R is the relative intensity of the
sound. Sounds that reach levels of 120 decibels or more are painful to humans. What is the
relative intensity of 120 decibels?
14. An earthquake rated at 3.5 on the Richter scale is felt by many people, and an
earthquake rated at 4.5 may cause local damage. The Richter scale magnitude
reading 𝑚 is given by 𝑚 = 𝑙𝑜𝑔10 𝑥, where 𝑥 represents the amplitude of the
seismic wave causing ground motion. How many times greater is the amplitude
of an earthquake that measures 4.5 on the Richter scale than on the measures
3.5?
15. Rob is solving a problem involving logarithms. He does everything correctly
except for one thing. He mistakenly writes 𝑙𝑜𝑔2 𝑎 + 𝑙𝑜𝑔2 𝑏 = 𝑙𝑜𝑔2 (𝑎 + 𝑏).
However, after substituting the values for 𝑎 and 𝑏 in his problem, he amazingly
still gets the right answer! The value of 𝑎 was 11. What must the value of 𝑏 have
been?