Change of Base Formula

Active learning helps students internalize the change of base formula by making the abstract relationship concrete. Converting logarithms between bases requires repeated use of the formula, so hands-on practice builds both accuracy and intuition. Collaborative work also allows students to test their understanding through immediate peer feedback.

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

Learning Objectives

1. Calculate the value of a logarithm with an arbitrary base using the change of base formula and base 10 or base e.
2. Compare the numerical results of evaluating the same logarithm using different bases (e.g., base 10 and base e) via the change of base formula.
3. Derive the change of base formula from the definition of a logarithm and properties of exponents.
4. Justify the selection of any positive real number (other than 1) as a valid base for the change of base formula.

Inquiry Circle

⏱ 20 min·Small Groups

Inquiry Circle: Deriving the Formula

Groups receive the steps of the change of base derivation in scrambled order and must arrange them in the correct sequence, then write the formula themselves. Each group presents their derivation to the class, explaining each algebraic step.

Prepare & details

Explain the utility of the change of base formula in evaluating logarithms.

Facilitation Tip: During Collaborative Investigation: Deriving the Formula, circulate and ask each group to share one step of their derivation before moving forward.

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

⏱ 15 min·Pairs

Think-Pair-Share: Base 10 vs. Base e

Pairs compute log_5(100) and log_3(50) using both log and ln versions of the change of base formula. They verify both methods give the same result and discuss which form they prefer and why. The class compares preferences and discusses when ln might be more useful.

Prepare & details

Compare the results of using different bases (e.g., 10 or e) in the change of base formula.

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

Think-Pair-Share

⏱ 20 min·Pairs

Individual Practice with Peer Check

Students independently evaluate four logarithms with non-standard bases using the change of base formula, showing all steps. Pairs exchange papers and verify each other's work, marking any steps where the base substitution was applied incorrectly.

Prepare & details

Justify why any base can be used in the change of base formula.

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

Teaching This Topic

Teach this topic by first anchoring the concept in calculator limitations, then moving to an algebraic derivation. Emphasize the importance of verifying results with base identities like log_b(b) = 1. Avoid rushing to the formula; let students discover it through guided exploration. Research shows that when students derive the formula themselves, they retain it longer and make fewer mechanical errors.

What to Expect

By the end of these activities, students will apply the change of base formula correctly in multiple contexts. They will explain why different bases yield the same numerical result and verify their work using simple test cases. Successful learners will also connect the formula to the broader concept of logarithmic equivalence.

Watch Out for These Misconceptions

Common Misconception

During Collaborative Investigation: Deriving the Formula, watch for students inverting the formula, writing log_b(x) = log(b) / log(x).

What to Teach Instead

Prompt the group to test their version with log_2(2). Ask them to compute both sides and compare to 1. When their version gives 1/1 instead of 1, guide them to switch numerator and denominator to match the identity.

Common Misconception

During Think-Pair-Share: Base 10 vs. Base e, watch for students assuming only base 10 or base e can be used in the denominator.

What to Teach Instead

Ask the pair to re-express log_3(9) using base 2 in the denominator. Have them calculate both log_10(9)/log_10(3) and log_2(9)/log_2(3) to see they arrive at the same value, then discuss why calculator availability, not mathematics, drives common practice.

Assessment Ideas

Quick Check

After Collaborative Investigation: Deriving the Formula, present students with log_4(64). Ask them to calculate its value using the change of base formula with base 10, then again with base e. Collect their work and verify they get 3 as the answer and show correct steps for both calculations.

Exit Ticket

During Individual Practice with Peer Check, give students the expression log_5(100). Ask them to write the expression using the change of base formula with base 10 and then with base e. Have them use a calculator to find the approximate value for both and confirm they are equal, writing one sentence about why this equality matters for using calculators.

Discussion Prompt

After Think-Pair-Share: Base 10 vs. Base e, pose the question: 'Why can we use any base in the change of base formula, and what does it mean if we get different numerical answers when using base 10 versus base e?' Facilitate a discussion where students explain the derivation and the concept of scalar multiples, listening for connections to logarithmic equivalence.

Extensions & Scaffolding

• Challenge: Ask students to create three different change of base expressions for log_7(50) using bases 2, 5, and 12, then compare results.
• Scaffolding: Provide a partially completed derivation table with missing steps for students to fill in during the Collaborative Investigation.
• Deeper exploration: Explore how the change of base formula relates to logarithmic functions being scalar multiples, using graphing technology to compare log_b(x) and log(x) for different bases b.

Key Vocabulary

Change of Base Formula	A formula that allows you to rewrite a logarithm in one base as a ratio of logarithms in another base, typically base 10 or base e.
Common Logarithm	A logarithm with a base of 10, often written as log(x) without an explicit base.
Natural Logarithm	A logarithm with a base of e (Euler's number), written as ln(x).
Logarithmic Equation	An equation that includes a logarithm. The change of base formula is used to solve or evaluate these when bases do not match calculator capabilities.