Kenneth ChengMathematical modeling
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LEARN · EXPLAIN · REVISE

Logarithms

Read the idea, work independently, then explain what changed.

高一必修 第一册(A版).pdf · 4.3 · PDF 129 / printed page 122

Revisit first: Exponential functions

TOPIC 01

Logarithms

Build understanding of logarithms through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Interpret and use logarithms with domain checks.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Inverse definition

A logarithm is the exponent needed to produce a positive argument.

log⁡ab=c  ⟺  ac=b\log_a b=c\iff a^c=b

Laws

Products become sums; quotients become differences; use only positive arguments.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in logarithms changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Exponential graph: horizontal shift 0, vertical shift 0. Input is replaced by x−0.

Exponential graph: horizontal shift 0, vertical shift 0. Input is replaced by x−0.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Worked example

Evaluate the logarithm.

log⁡24\log_24
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Rewrite logarithms as exponent equations and check base and argument conditions.
Hint 2
Ask which exponent produces the argument.
Worked solution

  1. Rewrite logarithms as exponent equations and check base and argument conditions.

  2. Calculate or simplify this relation.

    22=4⇒log⁡24=22^{2}=4\Rightarrow\log_24=2

  3. The logarithm is an exponent.

The requested value is 2.

Checks and common pitfalls: The logarithm is an exponent.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

Simplify using the product rule.

log⁡24+log⁡28\log_24+\log_2 8
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Rewrite logarithms as exponent equations and check base and argument conditions.
Hint 2
Adding logarithms multiplies positive arguments.
Worked solution

  1. Rewrite logarithms as exponent equations and check base and argument conditions.

  2. Calculate or simplify this relation.

    log⁡2(4⋅8)=log⁡232=5\log_2(4\cdot8)=\log_232=5

  3. It does not correspond to adding the arguments.

The requested value is 5.

Checks and common pitfalls: It does not correspond to adding the arguments.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

Solve the logarithmic equation.

log⁡2(x+2)=3\log_2(x+2)=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Rewrite logarithms as exponent equations and check base and argument conditions.
Hint 2
Convert to an exponential equation.
Worked solution

  1. Rewrite logarithms as exponent equations and check base and argument conditions.

  2. Calculate or simplify this relation.

    x+2=8⇒x=6;x+2=8>0x+2=8\Rightarrow x=6;\quad x+2=8>0

  3. Substitute the result into the positive-argument restriction.

The requested value is 6.

Checks and common pitfalls: Substitute the result into the positive-argument restriction.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Evaluate the logarithm.

log⁡28\log_28
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Rewrite logarithms as exponent equations and check base and argument conditions.
Hint 2
Ask which exponent produces the argument.
Worked solution

  1. Rewrite logarithms as exponent equations and check base and argument conditions.

  2. Calculate or simplify this relation.

    23=8⇒log⁡28=32^{3}=8\Rightarrow\log_28=3

  3. The logarithm is an exponent.

The requested value is 3.

Checks and common pitfalls: The logarithm is an exponent.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Simplify using the product rule.

log⁡28+log⁡28\log_28+\log_2 8
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Rewrite logarithms as exponent equations and check base and argument conditions.
Hint 2
Adding logarithms multiplies positive arguments.
Worked solution

  1. Rewrite logarithms as exponent equations and check base and argument conditions.

  2. Calculate or simplify this relation.

    log⁡2(8⋅8)=log⁡264=6\log_2(8\cdot8)=\log_264=6

  3. It does not correspond to adding the arguments.

The requested value is 6.

Checks and common pitfalls: It does not correspond to adding the arguments.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

Evaluate by change of base.

log⁡864\log_{8}64
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Rewrite logarithms as exponent equations and check base and argument conditions.
Hint 2
Use base two in numerator and denominator.
Worked solution

  1. Rewrite logarithms as exponent equations and check base and argument conditions.

  2. Calculate or simplify this relation.

    log⁡264log⁡28=63=2\frac{\log_264}{\log_28}=\frac{6}{3}=2

  3. A change of base needs the same new base in both places.

The requested value is 2.

Checks and common pitfalls: A change of base needs the same new base in both places.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

Find the real domain.

log⁡3(x−4)\log_3(x-4)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Rewrite logarithms as exponent equations and check base and argument conditions.
Hint 2
The argument must be strictly positive.
Worked solution

  1. Rewrite logarithms as exponent equations and check base and argument conditions.

  2. Calculate or simplify this relation.

    x−4>0x-4>0

  3. Zero is not an admissible logarithm argument.

x>4.

Checks and common pitfalls: Zero is not an admissible logarithm argument.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

Disprove the claimed addition rule using positive numbers.

log⁡2(a+b)=log⁡2a+log⁡2b\log_2(a+b)=\log_2a+\log_2b
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Rewrite logarithms as exponent equations and check base and argument conditions.
Hint 2
The valid rule uses a product inside the logarithm.
Worked solution

  1. Rewrite logarithms as exponent equations and check base and argument conditions.

  2. Calculate or simplify this relation.

    log⁡2(1+1)=1≠0=log⁡21+log⁡21\log_2(1+1)=1\ne0=\log_2 1+\log_2 1

  3. A logarithm does not distribute over addition.

Take a=b=1: the left side is 1 and the right side is 0.

Checks and common pitfalls: A logarithm does not distribute over addition.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

Evaluate the logarithm with a base below one.

log⁡1/39\log_{1/3}9
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Rewrite logarithms as exponent equations and check base and argument conditions.
Hint 2
A negative exponent turns the small base into a large value.
Worked solution

  1. Rewrite logarithms as exponent equations and check base and argument conditions.

  2. Calculate or simplify this relation.

    (1/3)−2=9(1/3)^{-2}=9

  3. Positive arguments can have negative logarithms.

The requested value is -2.

Checks and common pitfalls: Positive arguments can have negative logarithms.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

Solve the logarithmic equation.

log⁡2(x+3)=3\log_2(x+3)=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Rewrite logarithms as exponent equations and check base and argument conditions.
Hint 2
Convert to an exponential equation.
Worked solution

  1. Rewrite logarithms as exponent equations and check base and argument conditions.

  2. Calculate or simplify this relation.

    x+3=8⇒x=5;x+3=8>0x+3=8\Rightarrow x=5;\quad x+3=8>0

  3. Substitute the result into the positive-argument restriction.

The requested value is 5.

Checks and common pitfalls: Substitute the result into the positive-argument restriction.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

Evaluate the logarithm.

log⁡216\log_216
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Rewrite logarithms as exponent equations and check base and argument conditions.
Hint 2
Ask which exponent produces the argument.
Worked solution

  1. Rewrite logarithms as exponent equations and check base and argument conditions.

  2. Calculate or simplify this relation.

    24=16⇒log⁡216=42^{4}=16\Rightarrow\log_216=4

  3. The logarithm is an exponent.

The requested value is 4.

Checks and common pitfalls: The logarithm is an exponent.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

Simplify using the product rule.

log⁡216+log⁡28\log_216+\log_2 8
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Rewrite logarithms as exponent equations and check base and argument conditions.
Hint 2
Adding logarithms multiplies positive arguments.
Worked solution

  1. Rewrite logarithms as exponent equations and check base and argument conditions.

  2. Calculate or simplify this relation.

    log⁡2(16⋅8)=log⁡2128=7\log_2(16\cdot8)=\log_2128=7

  3. It does not correspond to adding the arguments.

The requested value is 7.

Checks and common pitfalls: It does not correspond to adding the arguments.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

Evaluate by change of base.

log⁡16256\log_{16}256
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Rewrite logarithms as exponent equations and check base and argument conditions.
Hint 2
Use base two in numerator and denominator.
Worked solution

  1. Rewrite logarithms as exponent equations and check base and argument conditions.

  2. Calculate or simplify this relation.

    log⁡2256log⁡216=84=2\frac{\log_2256}{\log_216}=\frac{8}{4}=2

  3. A change of base needs the same new base in both places.

The requested value is 2.

Checks and common pitfalls: A change of base needs the same new base in both places.

Think first. Reveal a hint when the class is ready.

Focus on one question

End-of-lesson check and correction

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    Choose a foundation skill to revisit ↗

    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for logarithms?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Interpret and use logarithms with domain checks.
    • Inverse definition: A logarithm is the exponent needed to produce a positive argument.
      log⁡ab=c  ⟺  ac=b\log_a b=c\iff a^c=b
    • Laws: Products become sums; quotients become differences; use only positive arguments.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: A logarithm is the exponent needed to produce a positive argument.
    • Expected reasoning: Products become sums; quotients become differences; use only positive arguments.
    • Expected correction: The logarithm of a sum does not split into a sum of logarithms.

    Assessment checklist

    • 1: identify the givens and required quantity.
    • 1: choose a valid definition, representation or method.
    • 1: present connected, correct reasoning.
    • 1: check conditions and explain the result.

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    Curriculum and source notes ↗