Exponentials and Natural Logarithms

Students often struggle with visualizing how exponential and logarithmic functions behave differently from polynomial or linear functions. Active learning through graphing, modeling, and comparison helps them internalize these relationships concretely. By engaging with multiple representations, students build intuition that supports later calculus work with derivatives and limits.

• Year 12
• Mathematics
• 4 activities
• 25 min–45 min

Learning Objectives

1. Explain the unique property of the number 'e' as the base where the derivative of e^x equals e^x.
2. Construct and analyze the graphical relationship between y = e^x and its inverse function y = ln(x), identifying key features.
3. Compare and contrast the growth patterns of exponential functions with linear functions, using specific examples.
4. Calculate solutions to equations involving exponential and natural logarithmic functions.

Problem-Based Learning

⏱ 30 min·Pairs

Pairs Graphing: Inverse Reflections

Pairs sketch y = e^x on graph paper, then reflect points over y = x to plot y = ln(x). They identify domain and range differences and test points like (0,1) and (1,0). Verify with class Desmos share.

Prepare & details

Explain what makes the number 'e' unique in the study of calculus.

Facilitation Tip: During Pairs Graphing: Inverse Reflections, circulate to ensure students test at least three points in Quadrant I and one near x = 0 to observe behavior.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Problem-Based Learning

⏱ 45 min·Small Groups

Small Groups: Growth Comparison Tables

Groups create tables comparing y = 2x and y = e^{0.5x} from x=0 to x=10. Plot points and discuss crossover where exponential overtakes linear. Extend to interpret doubling times.

Prepare & details

Construct the graph of y=e^x and y=ln(x) and analyze their relationship.

Facilitation Tip: For Growth Comparison Tables, ask groups to predict which function will dominate before plotting, then have them explain their reasoning aloud.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Problem-Based Learning

⏱ 40 min·Whole Class

Whole Class: Limit Demo for e

Project calculates (1 + 1/n)^n for increasing n values. Class predicts and votes on limit approaching e. Follow with pairs deriving derivative of e^x using definition.

Prepare & details

Compare the properties of exponential growth with linear growth.

Facilitation Tip: In the Limit Demo for e, pause after each step to let students predict what the next fraction will be, reinforcing the pattern.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Problem-Based Learning

⏱ 25 min·Individual

Individual: Log Equation Puzzles

Students solve 10 equations mixing e^x and ln(x), like ln(x) = 2 or e^{kx} = 5. Time themselves, then share strategies in plenary. Use for homework extension.

Prepare & details

Explain what makes the number 'e' unique in the study of calculus.

Facilitation Tip: During Log Equation Puzzles, remind students to check solutions by substituting back into the original equation to catch extraneous roots.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Teaching This Topic

Teachers should emphasize the inverse relationship between exponentials and logarithms before introducing e. Avoid starting with the derivative definition; instead, let students discover the special property of e through graphing and numerical exploration. Research shows that students grasp asymptotic behavior better when they first experience it visually through graphing before formal limits are introduced.

What to Expect

Students will demonstrate understanding by accurately graphing y = e^x and its inverse, comparing growth rates through tables, and explaining why e is significant in calculus. They will identify domain restrictions, asymptotic behavior, and the reflection property between the two functions.

Watch Out for These Misconceptions

Common Misconception

During Growth Comparison Tables, watch for students assuming exponential growth is always faster from the start.

What to Teach Instead

Have groups share their tables at the 5-minute mark and ask them to identify the crossover point where e^x surpasses 2x, then discuss why initial conditions matter in real-world contexts like interest rates.

Common Misconception

During Pairs Graphing: Inverse Reflections, watch for students testing negative values for ln(x).

What to Teach Instead

Prompt pairs to input x = -1 into their graphing software and observe the undefined behavior, then ask them to explain why the domain restriction x > 0 makes sense given y = e^x's output.

Common Misconception

During Limit Demo for e, watch for students treating e as just another base like 2 or 10.

What to Teach Instead

After the demo, ask students to compare the slope of y = e^x at x = 0 with the slopes of y = 2^x and y = 3^x at their respective y-intercepts to see the unique derivative property emerge.

Assessment Ideas

Quick Check

After Log Equation Puzzles, collect one sample from each student and check for correct isolation of x and explanation of each step, focusing on the use of ln to undo e.

Discussion Prompt

After Growth Comparison Tables, facilitate a whole-class discussion using the graphs of y = 2x and y = 2^x to explain linear versus exponential growth, then ask how e's properties change the conversation about rates of change.

Exit Ticket

During Pairs Graphing: Inverse Reflections, have students submit their labeled graphs before leaving, and check that they correctly identify the reflection line y = x and at least three matching points between the functions.

Extensions & Scaffolding

• Challenge: Ask students to find the derivative of y = e^(2x) and explain how the chain rule applies, connecting back to the original function's growth rate.
• Scaffolding: Provide pre-labeled axes for ln(x) with tick marks at y = -2, -1, 0, 1, 2 and ask students to plot y = e^x first to reference.
• Deeper exploration: Have students research the historical development of e and present how mathematicians like Euler approached its definition.

Key Vocabulary

Euler's number (e)	An irrational mathematical constant approximately equal to 2.71828. It is the base of the natural logarithm and is fundamental in calculus.
Natural exponential function	The function f(x) = e^x, where 'e' is Euler's number. Its derivative is itself, making it unique in calculus.
Natural logarithm	The inverse function of the natural exponential function, denoted as ln(x). It is the logarithm to the base 'e'.
Asymptote	A line that a curve approaches but never touches. For y = ln(x), the y-axis (x=0) is a vertical asymptote.