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Logarithms: The Inverse of Exponents

Eleventh graders learn logarithms — the inverse of exponentials — understanding that a logarithm asks “what exponent?” so log₂(8) = 3 because 2³ = 8, and using logs to solve exponential equations.

Grade 11Logarithms55 minutes1 class periodGradual ReleaseExplicit teaching4 StandardsCommon Core
Start the Lesson
Lesson at a Glance

Everything you need before the bell rings

Learning Objectives

Students will be able to…

  • ✓Understand a logarithm as an inverse.
  • ✓Convert between log and exponential form.
  • ✓Evaluate simple logs.
  • ✓Use logs to solve equations.
Essential Question

2 to WHAT power equals 8? That question has an answer — and a name: the logarithm. How do logs undo exponents, and why are they so powerful?

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Lesson Phases
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Vocabulary Terms
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Standards Aligned
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Interactive Task
Model · Interactive Tool

Log & Exponential Forms

Project this and change the base and exponent. Watch the logarithmic and exponential forms — they say the same thing two ways!

🔁 Convert between log and exponential formTry it
↕️ means the same as ↕️
Base b: 2
Exponent y: 3

The Lesson · Gradual Release (I Do · We Do · You Do)

55 minutes, five moves

Tap any phase to open the teacher moves and student actions.

1

Launch — 2 to What Power?

5 min

2 to WHAT power equals 8? That question has a name.

👩‍🏫 Teacher Moves

  • Ask the exponent question.
  • Introduce “logarithm.”
  • Set the goal: logarithms.

🎒 Student Actions

  • Consider the question.
  • Meet the log.
  • Predict the idea.
2

I Do — Log = Inverse of Exponent

10 min

Teacher models.

👩‍🏫 Teacher Moves

  • A log asks “what exponent?”
  • Convert log ↔ exponential.
  • Evaluate simple logs.

🎒 Student Actions

  • Learn the meaning.
  • Convert forms.
  • Evaluate logs.
3

We Do — Convert Together

13 min

Class uses Log & Exponential Forms.

👩‍🏫 Teacher Moves

  • Convert between forms.
  • Read both.
  • Explain the inverse.

🎒 Student Actions

  • Convert them.
  • Read both.
  • Explain it.
4

You Do — On Your Own

12 min

Students evaluate.

👩‍🏫 Teacher Moves

  • Convert log ↔ exponential.
  • Evaluate logs.
  • Solve exponential equations.

🎒 Student Actions

  • Convert them.
  • Evaluate them.
  • Solve some.
5

Close — Close

5 min

One log.

👩‍🏫 Teacher Moves

  • Give a log to evaluate.
  • Find its value.
  • Hand out the exit ticket.

🎒 Student Actions

  • Evaluate it.
  • Explain it.
  • Complete the exit ticket.
Standards Alignment

Built to the standards you report on

Aligned to the Common Core State Standards for Mathematics (High School Algebra II).

CCSS
F-LE.4

Express as a logarithm the solution to an exponential equation.

CCSS
F-BF.5

Understand the inverse relationship between exponents and logarithms.

CCSS
F-IF.7e

Graph exponential and logarithmic functions.

CCSS
A-SSE.3c

Use the properties of exponents.

Differentiation

One lesson, every learner

Multilingual Learners

ELL / EMERGING READERS
  • Log/exponential conversion cards.
  • Sentence frame: “logᵦ(x) = y because ___.”
  • Use a base-and-exponent chart.

Support & Access

IEP / 504
  • Start with base 2 and 10.
  • Convert to exponential form.
  • Evaluate step by step.

Stretch & Extend

GIFTED / EARLY FINISHERS
  • Use log properties.
  • Solve real exponential equations.
  • Graph a log function.
Materials

What to gather

  • 📽️Projector / board
  • 🧮Calculators
  • 💻Log & Exponential Forms
  • 📄Conversion charts
  • ✏️Pencils
  • 🎫Exit-ticket slips
Vocabulary

Key terms — hover for a quick definition

logarithmthe exponent a base is raised tobasethe number being raised to a powerexponentiala base raised to a variable powerinversean operation that undoes anotherargumentthe value inside the logcommon loga log with base 10natural loga log with base eexponenta power
Evaluate

Exit Ticket

Preview the three formative checks. Tap “Sample answer” to see what mastery looks like — hide them before you print for students.

QUESTION 1
What does log₃(9) equal?
2 (because 3² = 9).
QUESTION 2
Write 5² = 25 in logarithmic form.
log₅(25) = 2.
QUESTION 3
What operation does a logarithm undo?
Raising a base to a power (an exponent).

Going deeper? Log properties.

Have students explore the product and power rules of logarithms and use them to expand and condense log expressions. A printable logarithms sheet is in the Math library.

Study · Flashcards

Study the key terms

Tap a card to flip it, then rate whether you knew it. Built from this lesson’s vocabulary.

🃏 LogarithmsFlip
Card 1
Term
Tap to flip →
Meaning
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Practice · Quiz

Check your understanding

A quick self-check with instant feedback, drawn from this lesson’s key terms.

📝 LogarithmsQuiz
Score: 0
1 / 6
Question 1
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Practice · Worksheet

Printable worksheet

A print-and-go review sheet with a built-in answer key. Tap “Show answer key” to reveal answers, or print the clean version for students.

🖨️ LogarithmsPrint
Name: ________________________
Date: ____________

Part A · Write the word that matches each meaning

Word bank: argument, base, common log, exponent, exponential, inverse, logarithm, natural log
  1. the number being raised to a power
  2. a log with base e
  3. a log with base 10
  4. an operation that undoes another
  5. a base raised to a variable power
  6. the exponent a base is raised to
  7. the value inside the log
  8. a power

Part B · Show what you learned

  1. What does log₃(9) equal?
  2. Write 5² = 25 in logarithmic form.
  3. What operation does a logarithm undo?
Answer key — Part A: 1) base · 2) natural log · 3) common log · 4) inverse · 5) exponential · 6) logarithm · 7) argument · 8) exponent
Part B: 1) 2 (because 3² = 9). 2) log₅(25) = 2. 3) Raising a base to a power (an exponent).