Alg2 Lesson 3.2: Introduction to the Logarithm Function

Lesson 3.2: Introduction to the

Logarithm Function

Obj: I can formulate and solve exponential equations.

EQ: How do I find the inverse of an exponential
function?

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

Talk with your table group. Which role will you take on for this lesson?

• Facilitator

• Scribe

• Includer

• Resourcer

● Check off tasks & skills on calendar.

● Select skills to work on.

● Work on Deltamath.

Remember to work on the following too…

Part 1:

Investigating the Doubling Coin

​Read the story and express your thoughts.
A Payasam Story

Krishna in the form of an old sage, challenged the king of Ambalapuzzha to a game of chess. The prize, if won, would be one grain of rice on the first square of the chessboard, two on the second, four on the third, eight on the fourth, and so on, doubling the amount on the previous square. The king brashly agreed. Krishna, of course, won the game. The king started placing the rice grains and was shocked to see their number grow exponentially. By the end, he owed Krishna trillions of tons of rice! Amused at the king's confusion, Krishna revealed himself. "You don't have to give it all today." he said. "Just provide payasam to every Pilgrim who comes to my temple here, in search of comfort." Krishna's wish is honoured even today and payasam is served freely to all who visit the Ambalapuzzha Krishna temple.

Play Chess

Word Cloud

What are your thoughts about A Payasam Story?

The Doubling Coin - Investigation

Suppose that you have a coin whose value
doubles every day. That is, on the day you get
the coin, Day 0, its value is $1, on day 2, its
value is $2, on day 2, its value is $4, and so on,
doubling in value each additional day.

Would you prefer a Doubling Coin for 5
days or a single payment of $1,000,000?

Would you prefer a Doubling Coin for 10
days or a single payment of $1,000,000?

Would you prefer a Doubling Coin for 15
days or a single payment of $1,000,000?

The Doubling Coin - Part 1

Suppose that you have a coin whose
value doubles every day. That is, on
the day you get the coin, Day 0, its
value is $1, on day 2, its value is $2,
on day 2, its value is $4, and so on,
doubling in value each additional day.
How many days must you wait for the
coin to be worth $1,000,000?

1.

Write a function we can use to model the
value of the Doubling Coin. Justify the type
of function you selected for your model.

2.

Construct a graph of the function you
wrote in problem 1.

3.

How could we use this function model to
decide when to choose the Doubling Coin
over the single payment of $1,000,000?

Questions

How did you know when to choose the value of the Doubling Coin?

Using a table of values, on which day is the value of the Doubling Coin over the flat
amount of $1,000,000?

How can we use the graph to determine when the value of the Doubling Coin is greater
than $1,000,000?

How can you set up an equation that makes use of logarithms to solve the problem?

Drag and Drop

How did you know when to choose the value of the Doubling Coin? I used​

Placeholder

to determine that at ​

Placeholder

days, the value of the Doubling Coin was ​

Placeholder

than
$1,000,000.

Drag these tiles and drop them in the correct blank above

trial and error

20

30

15

greater

less

Drag and Drop

Using a table of values, on which day is the value of the Doubling Coin over the flat amount of $1,000,000? At ​

Placeholder

days, the value of the Doubling Coin is ​

Placeholder

than $1,000,000.

Drag these tiles and drop them in the correct blank above

30

15

less

20

more

Multiple Choice

How can we use the graph to determine when the value of the Doubling Coin is greater than $1,000,000?

• d = log10(1,000,000)
• d = log2(1,000,000)

Multiple Select

How can you set up an equation that makes use of logarithms to solve the problem?

• t = log₂(1,000,000)

• t = ln(1,000,000)

• 2^t =1,000,000

• t² = 1,000,000

• t = log(1,000,000)

Part 2:

Revisiting the Doubling Coin

The Doubling Coin Part 2

Let’s consider some specific values of the
Doubling Coin. In our discussion, we found that
we should wait 20 days for the coin to be worth
$1,000,000. How could we determine the
number of days before the Doubling Coin has a
value equal to any amount we choose? Work
through the following problems with your partner
so we can develop a function to make sense of
the Doubling Coin situation for any target
amount.

1.

If you were offered a single payment of $1,000 or a coin
that doubles for a specified number of days, for what
number of days should you opt for the Doubling Coin?

2.

If you were offered a single payment of $50,000 or a
coin that doubles for a specified number of days, for
what number of days should you opt for the Doubling
Coin?

3.

If you were offered a single payment of $125,000 or a
coin that doubles for a specified number of days, for
what number of days should you opt for the Doubling
Coin?

4.

How could you use a graph to solve problems 1,2, and
3?

5.

How could you set up an exponential equation and use
logarithms to solve problems 1,2, and 3?

6.

If we wanted to find the number of days for which the
Doubling Coin is worth any arbitrary amount of money
(let’s call it D dollars), how would we determine the
number of days we should wait until we opt for the
Doubling Coin?

Questions

What equations did you set up to determine the answers to problems 1,2, and 3?

What was the same and what was different about the equations you set up?

How did you rewrite each of these as logarithms to solve for t, the number of days it
takes to reach a specified constant value?

What was the same and what was different about the logarithms you set up?

Drag and Drop

The equations are:

1.) ​

Placeholder

2.) ​

Placeholder

3.) ​

Placeholder

Each of the equations included ​2^t, but each was equal to a different ​constant value.

Drag these tiles and drop them in the correct blank above

2^t=1,000

2^t=50,00

2^t=125,000

t=log₂(125,000)

t=log₂(50,000)

t=log₂(1,000)

Drag and Drop

To solve each, the logarithmic equations are

1.) ​

Placeholder

2) ​

Placeholder

3.) ​

Placeholder

Each of the logarithms included log, but each had a different argument.

Drag these tiles and drop them in the correct blank above

t=log₂(1,000)

t=log₂(50,000)

t=log₂(125,000)

2^t=125,000

2^t=1,000

2^t=50,000

Questions

What exponential equation could we set up to determine the number of days it
would take for the Doubling Coin to be worth D dollars? Use your work from the
previous problems to establish a pattern.

What logarithmic equation could we set up to determine the number of days it
would take for the Doubling Coin to be worth D dollars? Use your work from
previous problems to establish a pattern.

Multiple Choice

What exponential equation could we set up to determine the number of days, t, it would take for the Doubling Coin to be worth D dollars?

• 2^t=D

• 2^D=t

• D^t=2

• t^D=2

Multiple Choice

What logarithmic equation could we set up to determine the number of days, t, it would take for the Doubling Coin to be worth D dollars?

• t=log(D)

• D=log₂(t)

• t=log₂(D)

• D=log(t)

Part 3: Graphing Exponential

and Logarithmic Functions

Graphing Exponential and Logarithmic Functions

Use a graphing utility (desmos) the following functions and compare their features
and observe their properties.

Functions:

f(x)=2^x

g(x)=log_2(x)

Questions

What about the graphs of these functions
confirms that they are inverses?

What is the domain of f(x)=2^x and how does
that relate to the range of g(x)=log_2(x)?

What is the range of f(x)=2^x and how does that
relate to the domain of g(x)-log_2(x)?

The graph of f(x)=2^x increases very quickly.
How does this relate to the rate at which the
graph g(x)=log_2(x) grows?

Multiple Choice

What about the graphs of these functions
confirms that they are inverses?

• The functions are reflections across the line y=x.
• The functions are periodic and repeat every unit.
• The functions have the same rate of change at all points.

• They have inverse y-intercepts.

Drag and Drop

The domain of f(x) is ​

Placeholder

and the range of f(x) is ​

Placeholder

.

Since f(x) and g(x) are inverses, the domain of f(x) is ​

Placeholder

the range of g(x). Since the ​

Placeholder

-intercept of f(x) is (0, 1), the ​

Placeholder

-intercept of g(x) is (1, 0).

Drag these tiles and drop them in the correct blank above

all real numbers

(0, ∞)

the same as

different than

y

x

Multiple Choice

The graph of f(x)=2^x increases very quickly.
How does this relate to the rate at which the
graph g(x)=log_2(x) grows?

• The graph of g(x) = log_2(x) increases faster than f(x) = 2^x.

• The graph of g(x) = log_2(x) and f(x) = 2^x grow at the same rate.
• The graph of g(x) = log_2(x) decreases as x increases.
• The graph of g(x) = log_2(x) increases much slower than f(x) = 2^x.

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.