Alg2 Lesson 3.3: Log Rules & Transformations

Lesson 3.3: Connecting Properties of
Logarithms with Transformations of the
Graph of the Parent Logarithm Function

Obj: I can use log rules to expand and condense
logarithmic expressions and use them to identify
transformations.

EQ: How do I expand a logarithmic expression?

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

What is your decided role?

• Facilitator

• Scribe

• Resourcer

• Includer

● Check off tasks & skills on calendar.

● Select skills to work on.

● Work on Deltamath.

Remember to work on the following too…

Part 1: Exploring

Transformations of Logarithms

Desmos Activity Screen 1

https://student.desmos.com/join/eykq8x

Try to identify key features of the logarithmic function such as the
intercepts, the asymptotes, domain, range, and the shape of the
graph. Write these in your journal. Complete the questions on
here that follow after going through the first slide on the Desmos
Activity.

Multiple Choice

What do you notice about the x-intercepts of the graphs of f (x) = log_b(x) for any
value of the base, b?

• All logarithmic functions f(x) = log_b(x) do not have any x-intercepts.

• All logarithmic functions have an x-intercept at (0, 1).
• All logarithmic functions f(x) = log_b(x) have an x-intercept at (1, 0).
• The x-intercepts of logarithmic functions vary with the base.

Multiple Choice

Why does it make sense that the coordinate (1, 0) would be on all the logarithmic
graphs of the form log_b(x) you explored?

• The coordinate (1, 0) is on logarithmic graphs because log_b(1) = 1 for any nonzero b.
• The coordinate (1, 0) is on logarithmic graphs only for specific values of b.
• The coordinate (1, 0) is not on logarithmic graphs for b = 0.
• The coordinate (1, 0) is on all logarithmic graphs of the form log_b(x) because b^0 = 1 for any nonzero b.

• The coordinate (1, 0) corresponds to log_b(0) = 0 for any nonzero b.

Multiple Choice

Given y = log_b(x), why would a negative base not have an associated graph?

• A negative base does not have an associated graph because it leads to an inconsistent relationship between x and y.
• A negative base results in a linear relationship between x and y.
• Logarithmic functions are defined for all real numbers, including negative bases.
• A negative base can produce a continuous graph with oscillating values.

Multiple Choice

What did you notice about the shape (or behavior) of the graph of f(x) = log_b(x)
when b was greater than 1?

• The graph decreases as x increases and curves away from the y-axis as x approaches 0.
• The graph remains constant as x increases and has no curvature near the y-axis.
• The graph increases as x increases and curves toward the y-axis as x approaches 0.

Multiple Choice

What happened to the graph f(x) = log_b(x) when the base was a number between 0 and 1?

• The graph remains constant as x increases and does not curve toward the y-axis.
• The graph decreases as x increases and curves toward the y-axis as x approaches 0.
• The graph increases as x increases and curves away from the y-axis as x approaches 0.

Multiple Choice

What did you notice about the shape of the graph as the value of the base
increased?

• The greater the value of b, the “flatter” the graph of the function.

• The greater the value of b, the “steeper” the graph of the function.

Part 2: Exploring the

Product Rule

Desmos Activity Screen 2

https://student.desmos.com/join/eykq8x

Continue with the Desmos session from Part 1. On Screen 2, you and
your partner will investigate transformations of the function g(x) = log₂
(cx) by comparing it to the parent function f (x) = log₂(x). Slide the
green dot on the screen to change the value of c and observe the
resulting changes to the graph. The coordinates of two points of the
transformed function are shown on the graph. Make note of the values
of c that result in integer values for the coordinates of these points.

Drag and Drop

Consider the parent function f (x) = log_2(x). What kind of transformation would you expect the function g(x) = log_2(cx) to be? How would you expect the graphs to
differ? This is a ​

Placeholder

transformation applied to the input of the function, which should produce a ​

Placeholder

​

Placeholder

of the graph of the parent function.

Drag these tiles and drop them in the correct blank above

multiplicative

additive

horizontal

dilation

vertical

translation

reflection

What do you notice about the relationship among the
values of c, powers of 2, and the magnitude of the vertical
translation?

Multiple Choice

What do you notice about the values of c?

• The values of c are prime numbers.
• The values of c are factors of 2.

• The values of c are powers of 2.

Dropdown

Suppose that we wanted to express the powers of 2 using logarithms instead of
exponents. How would we express them? We can rewrite the expression 1=2^0

Select answer

, the expression 2=2^1 as ​

Select answer

, the expression 4=2^2 as ​

Select answer

, and the expression 8 = 2^3 as ​

Select answer

.

Examples to Add to your Journal

Part 3: Exploring the Power

Rule

Desmos Activity Screen 3

https://student.desmos.com/join/eykq8x

Continue with the Desmos session from Part 2. On Screen 3, you and
your partner will investigate transformations of the function
k(x) = log₂(x^d) by comparing it with the parent function f(x) = log₂(x). Slide the green dot on the screen to change the value of d and observe the resulting changes to the graph. As before, the coordinates of two points of the transformed function are shown on the graph. Make note of the values of d that result in integer values for the coordinates of these points.

Drag and Drop

Consider the parent function f (x) = log_2(x). What kind of transformation would you expect the function g(x) = log_2(x^d) to be? How would you expect the graphs to differ? This is a ​

Placeholder

transformation applied to the input of the function, which should produce a ​

Placeholder

​

Placeholder

of the graph of the parent function.

Drag these tiles and drop them in the correct blank above

multiplicative

additive

horizontal

dilation

vertical

translation

reflection

Write in your Journal

Drag and Drop

Suppose we had a logarithmic function whose argument was a base raised to a
negative exponent, such as k(x) = log_2(x^−1 ). If we use the power rule to rewrite

Placeholder

. This suggests that the transformation would be a ​

Placeholder

​

Placeholder

.

Drag these tiles and drop them in the correct blank above

−1•log_2(x)

log_2(x)-1

vertical

reflection

horizontal

translation

dilation

Drag and Drop

Suppose we had a logarithmic function whose argument was a base raised to a
negative exponent, such as k(x) = log_2(x^−2). If we use the product rule to rewrite

Placeholder

​

Placeholder

across the x-axis and a ​

Placeholder

​

Placeholder

.

Drag these tiles and drop them in the correct blank above

vertical

dilation

reflection

horizontal

translation

Part 4: Exploring the

Quotient Rule

Desmos Activity Screen 4

https://student.desmos.com/join/eykq8x

Continue with the Desmos session from Part 3. On Screen 4, you and your
partner will investigate transformations of the function h(x) = log₂(x/c) by
comparing it to the parent function f(x) = log₂(x). Slide the green dot on the
screen to change the value of c and observe the resulting changes to the
graph. As on the previous two screens, the coordinates of two points of the
transformed function are shown on the graph. Make note of the values of c
that result in integer values for the coordinates of these points.

Multiple Select

Consider a function h(x)=log_2(x/c), where c is a constant. How could we rewrite
the function so that the argument is a product?

• h(x)=log₂(x•c^-1)

• h(x)=log₂(1/c)•x

• h(x)=log₂(c/x)

• h(x)=log₂(x-c)

Multiple Choice

Expand $$h(x)=\log_2\left(\frac{x}{c}\right)$$

• $$h(x)=c\log_2\left(x\right)$$

• $$h(x)=\log_2\left(x\right)+\log_2\left(c\right)$$

• $$h(x)=\log_2\left(x\right)^c$$

• $$h(x)=\log_2\left(x\right)-\log_2\left(c\right)$$

Drag and Drop

What kind of transformation would you expect the graph of h(X)=log_2(x/c) to exhibit when you compare it to the parent function f(X)=log_2(x)? h will differ from f by a ​

Placeholder

​

Placeholder

.

Drag these tiles and drop them in the correct blank above

vertical

translation

reflection

horizontal

dilation

What do you notice about the relationship among the
values of c, powers of 2, and the magnitudes of the vertical
translation?

Write in your Journal

Multiple Select

Which of the following is equivalent to $$k(x)=\log_2(x^{-1})$$ ?

• $$k(x)=\log_2\left(\frac{1}{x}\right)$$

• $$k(x)=\log_2\left(x\right)-1$$

• $$k(x)=-\log_2\left(x\right)$$

• $$k(x)=\log_2\left(1\right)-\log_2\left(x\right)$$

Part 5: Handout 3.3: Using
Properties of Logarithms

Properties of Logarithms - Write these down!

Open Ended

Answer the essential question: How do I expand a logarithmic expression?

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.

Poll

What do you plan to do next?

• Deltamath

• Meditate on my notes

• Progress Quiz

• Redo these slides