Modeling with Exponential and Logarithmic Functions

Active learning works for this topic because students need to move between symbolic, graphical, and real-world representations to build deep understanding. These functions describe growth and decay patterns that are invisible until students manipulate and interpret them directly.

• 12th Grade
• Mathematics
• 4 activities
• 20 min–40 min

Learning Objectives

1. Analyze real-world data sets to determine if an exponential or logarithmic model is most appropriate.
2. Construct exponential and logarithmic functions to accurately model population growth and radioactive decay scenarios.
3. Evaluate the impact of changing parameters (e.g., growth rate, initial value) on the behavior of exponential and logarithmic models.
4. Compare and contrast the characteristics of exponential growth versus exponential decay functions, identifying the role of the base.
5. Explain the relationship between exponential and logarithmic functions and their application in compound interest calculations.

Think-Pair-Share

⏱ 25 min·Pairs

Think-Pair-Share: Choosing the Right Model

Pairs receive four data sets printed on cards: one linear growth, one exponential growth, one exponential decay, and one logarithmic growth. Partners graph each set by hand or on a calculator, identify the model type, and write one sentence justifying their classification. They then share their reasoning with another pair before a whole-class discussion on distinguishing features.

Prepare & details

How can exponential functions be used to model situations involving constant percentage change?

Facilitation Tip: During the Think-Pair-Share, circulate and listen for students to verbalize why they rule out polynomial or linear models before sharing with the class.

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

⏱ 40 min·Small Groups

Inquiry Circle: Half-Life Lab

Groups model radioactive decay by flipping coins: each flip represents one half-life, and any coin showing tails is decayed and removed. Groups record the number of remaining coins each round and plot the decay curve. They then fit an exponential model to their data and compare their experimental base to the theoretical 0.5 per half-life.

Prepare & details

Analyze the relationship between exponential growth/decay and the base of the exponential function.

Facilitation Tip: During the Collaborative Investigation, ask groups to predict how many half-lives remain after each step before they calculate to build intuition.

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: Parameter Interpretation

Stations display four exponential or logarithmic models from different real contexts: population data, COVID case counts, compound interest, and pH scale. Groups visit each station and write an interpretation of each model's base and initial value in plain language. At the final station, groups predict the model's output ten units beyond the data range and justify their estimate.

Prepare & details

Construct an exponential or logarithmic model to fit a given set of data.

Facilitation Tip: During the Gallery Walk, direct students to focus first on the parameters' meaning before comparing numerical values.

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

Case Study Analysis

⏱ 20 min·Individual

Individual Practice: Fit and Justify

Students receive a table of data from a real demographic source, such as US Census population data for a specific state. Individually they determine whether a linear or exponential model fits better, write the model equation, and write two sentences interpreting the growth rate in context.

Prepare & details

How can exponential functions be used to model situations involving constant percentage change?

Facilitation Tip: During the Individual Practice, require students to sketch a rough graph before writing the function to catch scaling errors early.

Setup: Groups at tables with case materials

Materials: Case study packet (3-5 pages), Analysis framework worksheet, Presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Teaching This Topic

Start with concrete data sets so students see the actual curves rather than abstract functions. Use side-by-side plots to contrast exponential and polynomial growth visually. Require written justifications for every model choice to prevent guessing. Avoid rushing to formulas; insist on the inverse relationship between logs and exponents until students can derive one from the other.

What to Expect

Successful learning looks like students choosing the correct model for a scenario, interpreting parameters in context, and fluently converting between exponential and logarithmic forms. They should justify their choices with rate-of-change reasoning and correct use of inverse relationships.

Watch Out for These Misconceptions

Common Misconception

During Think-Pair-Share, watch for students stating 'the larger the base, the faster it grows' without checking base size.

What to Teach Instead

After plotting two exponential functions with the same initial value but bases 2 and 0.5, ask groups to describe how each function behaves over time and why the base direction matters.

Common Misconception

During Gallery Walk, watch for students treating logarithmic and exponential functions as unrelated.

What to Teach Instead

During the log-exponential form matching portion of the walk, have students convert each card to the other form and verify by evaluating both expressions.

Common Misconception

During Collaborative Investigation, watch for students assuming all curved-up data is exponential.

What to Teach Instead

Before modeling, have groups compute first and second differences in their data tables to confirm proportional growth rates typical of exponential models.

Assessment Ideas

Quick Check

After Think-Pair-Share, present two short data sets and ask students to identify which is exponential and which is linear, explaining their reasoning based on rate of change.

Discussion Prompt

During Gallery Walk, ask students to discuss when a logarithmic model might fit better than an exponential one, using examples like sound intensity or earthquake magnitude to justify their choices.

Exit Ticket

After Individual Practice, give each student a card with a scenario and ask them to write the function type, key parameter, and a brief justification for their choice.

Extensions & Scaffolding

• Challenge early finishers to create their own data set that fits an exponential decay model and present it to peers for fitting.
• Scaffolding for struggling students: Provide a partially completed table of values with first differences already computed to guide model selection.
• Deeper exploration: Have students research a real-world scenario (e.g., CO2 levels, bacteria growth) and present both the model and its limitations in context.

Key Vocabulary

Exponential Growth	A pattern where a quantity increases at a rate proportional to its current value, resulting in a J-shaped curve.
Exponential Decay	A pattern where a quantity decreases at a rate proportional to its current value, resulting in a curve that approaches zero.
Logarithmic Function	The inverse of an exponential function, used to model phenomena that grow or decay very rapidly initially and then slow down.
Half-life	The time required for a quantity of a substance undergoing exponential decay to reduce to half of its initial value.
Compound Interest	Interest calculated on the initial principal and also on the accumulated interest from previous periods.