Introduction to Exponential Functions

Active learning works for exponential functions because the abstract nature of logarithms requires students to move beyond symbolic manipulation. When students translate between exponential and logarithmic forms, manipulate physical representations, and investigate properties collaboratively, they build durable mental models of how these functions relate to one another.

• 11th Grade
• Mathematics
• 3 activities
• 15 min–35 min

Learning Objectives

1. Define an exponential function and identify its key components, including the base and initial value.
2. Graph exponential growth and decay functions, accurately plotting points and indicating the general shape of the curve.
3. Identify the y-intercept and horizontal asymptote of an exponential function from its equation and graph.
4. Compare the rate of change of exponential functions to linear and polynomial functions using graphical and numerical methods.
5. Analyze how changes to the base of an exponential function affect its rate of growth or decay.

Think-Pair-Share

⏱ 15 min·Pairs

Think-Pair-Share: The Log Translation

Students are given a list of exponential equations and must work with a partner to translate them into logarithmic form and vice versa. They discuss why the base remains the same in both versions.

Prepare & details

Explain the defining characteristics of an exponential function.

Facilitation Tip: During The Log Translation, ask students to verbalize the connection between each written step and the corresponding exponential form to reinforce the inverse relationship.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Inquiry Circle

⏱ 35 min·Small Groups

Inquiry Circle: Log Property Discovery

Groups use calculators to find the logs of various numbers and look for patterns. For example, they might find that log(2) + log(3) equals log(6), leading them to 'discover' the product rule on their own.

Prepare & details

Compare the growth rate of exponential functions to linear and polynomial functions.

Facilitation Tip: In Log Property Discovery, circulate and look for groups that test edge cases, such as log(1) or log(0), to expand their understanding beyond standard examples.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Gallery Walk

⏱ 30 min·Small Groups

Gallery Walk: Logarithmic Scales

Post examples of the Richter scale, pH scale, and decibel scale around the room. Students move in groups to explain how logarithms allow these scales to represent huge differences in intensity using small numbers.

Prepare & details

Analyze the role of the base in determining the growth or decay of an exponential function.

Facilitation Tip: During the Gallery Walk of Logarithmic Scales, encourage students to explain how each scale (e.g., Richter, pH) reflects the underlying logarithmic growth pattern in real-world contexts.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Teaching This Topic

Teachers approach this topic by first establishing the inverse relationship between exponents and logarithms through concrete examples. Avoid starting with formal definitions; instead, let students discover properties by testing values and observing patterns in collaborative settings. Research shows that when students physically rearrange equation components, such as moving the exponent to the other side of a log expression, their retention of the concept improves significantly.

What to Expect

Successful learning looks like students explaining why log(A + B) is not equal to log(A) + log(B), confidently applying log properties to simplify expressions, and accurately solving exponential equations by converting between forms. Students should articulate that a logarithm is simply an unknown exponent and use this understanding to justify each step in their work.

Watch Out for These Misconceptions

Common Misconception

During Collaborative Investigation: Log Property Discovery, watch for students who incorrectly generalize that log(A + B) equals log(A) + log(B).

What to Teach Instead

Prompt these students to test their hypothesis with a calculator during the investigation, then guide them to notice that log(A + B) is not equal to log(A) + log(B) by comparing numerical values. Reinforce that the product rule only applies when the arguments are multiplied, not added.

Common Misconception

During Think-Pair-Share: The Log Translation, watch for students who confuse the roles of the base and the exponent in logarithmic expressions.

What to Teach Instead

Use the Human Equation activity structure here: have students physically stand on either side of a log sign to represent the base, the argument, and the result. By moving the 'exponent' student to the other side of the log sign, they can see how the value they’re solving for becomes the exponent in the exponential form.

Assessment Ideas

Quick Check

After Think-Pair-Share: The Log Translation, present students with several function equations (e.g., y = 3(2)^x, y = 5(0.5)^x, y = x^2). Ask them to identify which are exponential and explain their reasoning based on the definition. Circulate to listen for students who correctly identify the exponential functions by their form and articulate why others are not exponential.

Exit Ticket

After Collaborative Investigation: Log Property Discovery, ask students to write a one-sentence explanation of why log_b(MN) = log_b(M) + log_b(N) is true, using at least one numerical example to support their reasoning. Collect these to assess their understanding of the product rule.

Discussion Prompt

During Gallery Walk: Logarithmic Scales, pose the question: 'How does the graph of y = 2^x differ from the graph of y = 10^x?' Facilitate a discussion where students compare the steepness of the curves and the role of the base in their growth rates, using their observations from the gallery walk as evidence.

Extensions & Scaffolding

• Challenge early finishers to create their own exponential/logarithmic equation and challenge a partner to solve it, justifying each step using the properties they’ve learned.
• For students who struggle, provide a scaffolded worksheet with partially filled-in steps for applying log properties to equations, focusing on one rule at a time.
• Deeper exploration: Have students research how logarithms are used in measuring sound intensity (decibels) or earthquake magnitudes (Richter scale) and present their findings to the class.

Key Vocabulary

Exponential Function	A function of the form f(x) = ab^x, where 'a' is the initial value and 'b' is the positive base (b ≠ 1), representing rapid growth or decay.
Base (b)	The constant factor by which the variable quantity is multiplied in each step of an exponential function; determines growth (b > 1) or decay (0 < b < 1).
Growth Factor	The base of an exponential function when it is greater than 1, indicating that the function's value increases over time.
Decay Factor	The base of an exponential function when it is between 0 and 1, indicating that the function's value decreases over time.
Horizontal Asymptote	A horizontal line that the graph of an exponential function approaches but never touches, typically y = 0 for basic exponential functions.