The Number 'e' and Natural Logarithms

Active learning helps students grasp the abstract nature of e and natural logarithms by connecting them to concrete experiences. Working with limits, inverses, and real-world growth models makes these concepts less intimidating and more intuitive.

• 11th Grade
• Mathematics
• 3 activities
• 20 min–25 min

Learning Objectives

1. Calculate the value of the mathematical constant 'e' by evaluating the limit of (1 + 1/n)^n as n approaches infinity.
2. Explain the relationship between the natural exponential function, f(x) = e^x, and its inverse, the natural logarithmic function, f(x) = ln(x).
3. Analyze real-world scenarios, such as population growth or radioactive decay, and model them using exponential functions with base 'e'.
4. Compare the growth rates of exponential functions with different bases, identifying 'e' as the base for continuous growth.

Inquiry Circle

⏱ 25 min·Small Groups

Inquiry Circle: Discovering e

Groups compute (1 + 1/n)^n for n = 1, 10, 100, 1000, and 10000 using calculators. They record how the output approaches a limit, then share observations with the class and the teacher reveals the connection to continuous compounding.

Prepare & details

Justify the significance of the number 'e' in continuous growth models.

Facilitation Tip: During Gallery Walk: ln and e as Inverses, quietly note which groups still confuse ln and log, and plan a brief mini-lesson for the next class.

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

Think-Pair-Share

⏱ 20 min·Pairs

Think-Pair-Share: Where Does e Appear?

Provide pairs with a list of formulas from physics, biology, finance, and engineering. Each pair identifies which formulas use e, explains why continuous change requires this base, and presents one example to the class.

Prepare & details

Explain the relationship between the natural logarithm and the exponential function with base 'e'.

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

Gallery Walk

⏱ 25 min·Small Groups

Gallery Walk: ln and e as Inverses

Post four stations showing graphs and tables of e^x and ln(x) with different scales. Groups annotate each poster, marking the inverse relationship, identifying domain and range, and explaining why ln(e^x) = x and e^(ln x) = x.

Prepare & details

Analyze real-world phenomena where 'e' naturally appears.

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

Teach e by starting with compound interest and letting students derive the continuous model themselves. Avoid rushing to the definition through limits first, as this can obscure the intuitive foundation. Research shows that multiple representations—numerical, graphical, and contextual—build deeper understanding of e and ln than symbolic manipulation alone.

What to Expect

Students will confidently recognize e as a fixed constant, explain why ln is the inverse of e^x, and apply these ideas to continuous growth scenarios. Success looks like accurate computation, clear explanations, and correct use of notation.

Watch Out for These Misconceptions

Common Misconception

During Collaborative Investigation: Discovering e, watch for students who treat e as a variable and write expressions like e = 2.718 in their calculations.

What to Teach Instead

Circulate and ask each group: 'Is 2.718 an exact value or an approximation?' Then have them replace it with the limit expression (1 + 1/n)^n and explain why e is fixed, not variable.

Common Misconception

During Gallery Walk: ln and e as Inverses, watch for students who confuse ln(x) with log(x) and assume they are interchangeable.

What to Teach Instead

Provide a side-by-side table where groups compute ln(100), log(100), e^4.605, and 10^4.605. Ask them to compare the outputs and observe the constant multiplier between ln and log for the same input.

Assessment Ideas

Quick Check

After Collaborative Investigation: Discovering e, ask students to rewrite the compound interest formula A = P(1 + r/n)^(nt) to represent continuous compounding using Euler’s number e. Collect responses to check if they correctly write A = Pe^(rt).

Exit Ticket

After Think-Pair-Share: Where Does e Appear?, give students two functions: f(x) = 2^x and g(x) = e^x. Ask them to explain in 1-2 sentences which function represents continuous growth and why. Collect and review for accurate reasoning and the correct approximation of e.

Discussion Prompt

During Gallery Walk: ln and e as Inverses, pose the question: 'Why is the number e considered the natural base for exponential functions?' Facilitate a discussion where students connect e to its derivative property and its appearance in continuous growth models. Listen for connections between e and exponential growth or decay.

Extensions & Scaffolding

• Challenge students to find the exact value of e to five decimal places by graphing y = (1 + 1/x)^x and using the calculator’s zoom feature.
• Scaffolding: Provide a partially completed table for students to fill in values of (1 + 1/n)^n for n = 1, 2, 5, 10, 100, 1000.
• Deeper exploration: Have students research how e appears in the normal distribution and prepare a short explanation or poster.

Key Vocabulary

Euler's number (e)	An irrational mathematical constant, approximately 2.71828, that is the base of the natural logarithm and arises naturally in calculus and compound interest.
Natural exponential function	A function of the form f(x) = e^x, where 'e' is Euler's number. Its unique property is that its derivative is itself.
Natural logarithm (ln)	The inverse function of the natural exponential function, denoted as ln(x). It answers the question: 'To what power must e be raised to get x?'
Continuous growth	A model of growth where the rate of increase is proportional to the current amount, leading to exponential growth described by functions involving 'e'.