Alg2 Lesson 3.4: Applications of Logarithms

Lesson 3.4: Applications of

Logarithms

Obj: I can formulate and solve logarithmic and
exponential equations.

EQ: How do I solve logarithmic equations given a
real world scenario?

Roles:
Facilitator
Scribe
Resourcer
Includer

Lesson Goals:
● Creative Thinking
● Talk through controversies and conflict
● Recognize and reduce ambiguity
● Encourage thinking based on formulas and prior info
● Help explain ideas to each other
● Own your ideas and work
● Record ideas in your journal
● Answer Questions on Slides
● Follow your team roles

Facilitator

• Make sure that all peers are staying on task.

• Give advice or suggestions to resolve the problem.

• Be sure everyone is able to explain.

Scribe

• Make sure peers organize their results on their own papers.

• Remind peers to use color, arrows, and other math tools to
communicate your mathematics, reasons, and connections.

• Be ready to join the teacher for a huddle.

Resourcer

• Make sure peers are getting the materials needed.

• Make sure that all materials are put away neatly.

• Make sure that peers are logged in to the needed site.

• Help troubleshoot any technology difficulties that may arise.

Includer

• Make sure that all peers are talking about their work.

• Helps keep peers’ voice volume low.

• Encourages everyone to ask questions.

• Communicates conflicts or questions to the teacher.

Poll

How do you plan to do with your group?

• Redirect them to complete the slides.

• Compliment their journal notes.

• Ask them questions.

• Try to get a higher score than them on the first try.

● Check off tasks & skills on calendar.

● Select skills to work on.

● Work on Deltamath.

Remember to work on the following too…

Part 1: Exploring the Utility

of Logarithms

Exploring the Utility of Logarithms

What do you notice and what do you wonder
about the graph?

What kind of function would you use to model
the data? What evidence do you have to support
your claim?

What kind of function might you use to model the
data?

Questions

What do you notice about the values on the horizontal axis?

What do you notice about the values on the vertical axis?

What does this mean about the distance between the endpoints of the intervals
indicated on the vertical axis?

What kind of function have we used to model data where the output increases by a factor instead of a constant amount?

Do you think an exponential function could be a good model for the data displayed on the scatterplot? Why or why not?

Multiple Choice

What do you notice about the values on the horizontal axis?

• They are the number of transistors, but their labels do not increase by a constant increment.

• They are the years starting in 1970. They are labeled in increments of two.

Multiple Choice

What kind of function is used to model data where the output increases by a factor instead of a constant amount?

• Exponential

• Linear

• Piecewise

• Factorial

Exploring the Utility of Logarithms

Here is the same date set but the vertical axis is
labelled in constant increments of 5 billion.

Do you think an exponential function would be a
good model for this data set? Why?

Exploring the Utility of Logarithms

The vertical axis on this graph is scaled logarithmically
rather than linearly.

This kind of graph is called a semi-log graph because
one axis uses powers of 10 rather than equal distances
between the values

A semi-log graph can be a very effective way to express
large numbers because it has the effect of presenting the
data linearly rather than exponentially.

-

This way of using logarithms to present data more
manageably is one of the common uses of
logarithms in modern science

Multiple Choice

Do you think an exponential function could be a good model for the data displayed on the scatterplot?

• Yes

• No

Part 2: Linearizing
Exponential Data

Linearizing Exponential Data

We want to explore how quickly the output
values of an exponential function get too large to
easily graph using constant scaling on both
axes.

Using the function you are assigned, create a
table of values for integer inputs of −2 through 8.
Then graph the function on graph paper.

Questions

What do you notice about the output values of the function?

How will this rapid increase in the output values affect the graph that you made?

Multiple Choice

What do you notice about the output values of the function?

• The output values are decreasing steadily.
• The output values remain constant over time.
• The output values are quickly getting very large.

Linearizing Exponential Data

You can use logarithms to “tame” the
rapid growth you observe in your
output values.

Add a 3rd column to your table of
values. In this 3rd column take the
natural logarithm of each output
value (round to 3 decimal places).
Then plot the points on graph paper.

Questions

What do you notice about this graph? How is it different from the first graph?

Why do you think that taking the natural logarithm of the y-values transformed the
appearance of the graph from exponential to linear?

Multiple Choice

What do you notice about the natural log of f(x)? How is it different from f(x)?

• This graph appears to be linear. f(x) was exponential.

• This graph appears to be exponential. f(x) was linear.

• This graph appears to be linear. f(x) was logarithmic.

Multiple Choice

Why do you think that taking the natural logarithm of the y-values transformed the appearance of the graph from exponential to linear?

• Taking the natural logarithm has no effect on the graph's appearance.
• The natural logarithm makes linear data appear exponential.
• The logarithm transforms the exponential data into a linear form.

Linearizing Exponential Data

Use a graphing utility (desmos) to determine a linear regression equation for the
x-values and the natural logarithm of the y-values. Then plot the residuals.

The laws of logarithms can be used to demonstrate that taking the logarithm of
an exponential fruition yields a linear function.

Linearizing Exponential Data

Write the transformed function algebraically.

f(x)=3(2)^x becomes g(x)=ln3(2^x)

Use the laws of logarithms to rewrite this function g in an equivalent form.

-

Graph g looks linear, so write in the form of a linear function)

Linearizing Exponential Data

Find the decimal approximation for each logarithmic expression in your new
function.

Main Points

●

The key property of logarithms is that they turn large or small numbers into
more reasonable quantities that are easier to analyze.

○

This is often used in scientific concepts so that very large numbers can be expressed more
concisely.

●

Real World Examples of Logarithmic Scales Include:

○

Decibel scale

○

pH scale

○

Richter scale

●

Logarithms are a powerful tool for expressing large numbers as smaller, more
manageable quantities. They are also used to turn small numbers, such as
those expressed in scientific notation, into more reasonable quantities.

Part 3: Understanding the

Richter Scale

Understanding the Richter Scale

The Richter scale quantifies the magnitude of an earthquake by comparing the relative intensity of a given earthquake to the smallest measurable earthquake. The Richter scale uses logarithms to express the wide-ranging intensities with smaller numbers. This example also provides student with a chance to see how he laws of logarithms can be used to solve applied mathematics problems.

Understanding the Richter Scale

On Handout 3.4A: Exploring the Richter Scale you will examine facts about 2
earthquakes that occurred in Chile and California

First we will learn how the Richter scale works in this video.

After the video:

-

Work on Handout 3.4A

-

Make sure you use properties of logs
to answer the questions

Handout 3.4 a Question #6

The 1906 San Francisco earthquake was one of the most destructive and infamous natural disasters in California’s history. Though the San Francisco earthquake was powerful, the Valdivia earthquake was 50 times more powerful. Use this fact to determine the magnitude of the 1906 San Francisco earthquake.

Handout 3.4 b Question #3

Sounds above 85 dB are generally considered unsafe, and sustained exposure to these sounds can cause hearing damage over time. How many times louder than I₀ is a sound that measures 85 dB?

​Note: Max volume on most headphones is about 100 dB.
Hearing Loss Article: https://www.healthyhearing.com/report/41775-Degrees-of-hearing-loss

Random Question of the Day Time

https://wheelofnames.com/4ke-epz We’ll spin the
wheel as a class and spend a minute or so
discussing our answers.