Kenneth ChengMathematical modeling
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Exponential and logarithmic equations

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

2027 JM01 考試大綱 · 8. Logarithmic and exponential functions · PDF 2 / printed page 2

Revisit first: ExponentsLogarithmsLogarithmic functions

TOPIC 01

Exponential and logarithmic equations

Build understanding of exponential and logarithmic equations through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Choose an equivalent method and explain exclusions before calculating.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Positive arguments

A real logarithm requires a positive argument and a positive base unequal to one.

Substitution

Exponential quadratic substitution produces a positive variable; logarithm laws retain the original domain.

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 exponential and logarithmic equations 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

Solve the exponential equation.

2x+1=252^{x+1}=2^{5}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.
Hint 2
The base 2 exponential is injective.
Worked solution

  1. Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.

  2. Calculate or simplify this relation.

    x+1=5x+1=5

  3. Compare exponents only after putting both sides over the same base.

The requested value is 4.

Checks and common pitfalls: Compare exponents only after putting both sides over the same base.

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

02 / Standard#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
Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.
Hint 2
Rewrite in exponential form.
Worked solution

  1. Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.

  2. Calculate or simplify this relation.

    x>2;x−2=8x>2;\quad x-2=8

  3. The resulting root must keep the argument positive.

The requested value is 10.

Checks and common pitfalls: The resulting root must keep the argument positive.

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

03 / Transfer#Worked example

Use x=−2 to explain why log(x²)=2 log(x) is not an identity for every nonzero real x.

x=−2;log⁡2(x2)=2log⁡2xx=-2;\quad \log_2(x^2)=2\log_2 x
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.
Hint 2
Check the domain of each side before using a log law.
Worked solution

  1. Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.

  2. Calculate or simplify this relation.

    log⁡2(x2)=2log⁡2∣x∣(x≠0)\log_2(x^2)=2\log_2|x|\quad(x\ne0)

  3. An algebraically plausible log law may change the domain.

For x<0 the right side is undefined; use 2 log₂|x| for x≠0.

Checks and common pitfalls: An algebraically plausible log law may change the domain.

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.

04 / Foundation#Your turn

Solve the exponential equation.

2x+1=262^{x+1}=2^{6}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.
Hint 2
The base 2 exponential is injective.
Worked solution

  1. Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.

  2. Calculate or simplify this relation.

    x+1=6x+1=6

  3. Compare exponents only after putting both sides over the same base.

The requested value is 5.

Checks and common pitfalls: Compare exponents only after putting both sides over the same base.

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

05 / Foundation#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
Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.
Hint 2
Rewrite in exponential form.
Worked solution

  1. Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.

  2. Calculate or simplify this relation.

    x>3;x−3=8x>3;\quad x-3=8

  3. The resulting root must keep the argument positive.

The requested value is 11.

Checks and common pitfalls: The resulting root must keep the argument positive.

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

06 / Foundation#Your turn

Solve the equation, with the domain restriction.

log⁡2(x−3)+log⁡2(x−4)=1\log_2(x-3)+\log_2(x-4)=1
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.
Hint 2
Set t=x−3; then t(t−1)=2.
Worked solution

  1. Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.

  2. Calculate or simplify this relation.

    x>4;(x−3)(x−4)=2;x=5 or 2x>4;\quad (x-3)(x-4)=2;\quad x=5\text{ or }2

  3. The smaller algebraic root makes both logarithm arguments negative.

The requested value is 5.

Checks and common pitfalls: The smaller algebraic root makes both logarithm arguments negative.

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

07 / Foundation#Your turn

Solve the exponential quadratic completely.

4x−4⋅2x+3=04^x-4\cdot2^x+3=0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.
Hint 2
Treat 4ˣ as (2ˣ)².
Worked solution

  1. Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.

  2. Calculate or simplify this relation.

    t=2x>0;(t−1)(t−3)=0t=2^x>0;\quad(t-1)(t-3)=0

  3. The substitution variable must be positive; both roots here are positive.

x=0 or log₂(3).

Checks and common pitfalls: The substitution variable must be positive; both roots here are positive.

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

Solve the logarithmic inequality with a base below one.

log⁡1/2x>3\log_{1/2}x>3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.
Hint 2
The base 1/2 logarithm is decreasing.
Worked solution

  1. Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.

  2. Calculate or simplify this relation.

    x>0;x<(1/2)3x>0;\quad x<(1/2)^{3}

  3. Reverse the order when converting the logarithmic comparison.

0<x<1/8.

Checks and common pitfalls: Reverse the order when converting the logarithmic comparison.

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

A population doubles every 3 years. How many years are needed for an eightfold increase?

N(t)=N0 2t/3;N0>0N(t)=N_0\,2^{t/3};\quad N_0>0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.
Hint 2
Eightfold means three doublings.
Worked solution

  1. Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.

  2. Calculate or simplify this relation.

    2t/3=8=23;t=92^{t/3}=8=2^3;\quad t=9

  3. The model assumes the doubling rate remains constant.

The requested value is 9.

Checks and common pitfalls: The model assumes the doubling rate remains constant.

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

10 / Standard#Your turn

Use x=−3 to explain why log(x²)=2 log(x) is not an identity for every nonzero real x.

x=−3;log⁡2(x2)=2log⁡2xx=-3;\quad \log_2(x^2)=2\log_2 x
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.
Hint 2
Check the domain of each side before using a log law.
Worked solution

  1. Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.

  2. Calculate or simplify this relation.

    log⁡2(x2)=2log⁡2∣x∣(x≠0)\log_2(x^2)=2\log_2|x|\quad(x\ne0)

  3. An algebraically plausible log law may change the domain.

For x<0 the right side is undefined; use 2 log₂|x| for x≠0.

Checks and common pitfalls: An algebraically plausible log law may change the domain.

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.

11 / Standard#Your turn

Solve the exponential equation.

2x+1=272^{x+1}=2^{7}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.
Hint 2
The base 2 exponential is injective.
Worked solution

  1. Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.

  2. Calculate or simplify this relation.

    x+1=7x+1=7

  3. Compare exponents only after putting both sides over the same base.

The requested value is 6.

Checks and common pitfalls: Compare exponents only after putting both sides over the same base.

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

12 / Transfer#Your turn

Solve the logarithmic equation.

log⁡2(x−4)=3\log_2(x-4)=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.
Hint 2
Rewrite in exponential form.
Worked solution

  1. Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.

  2. Calculate or simplify this relation.

    x>4;x−4=8x>4;\quad x-4=8

  3. The resulting root must keep the argument positive.

The requested value is 12.

Checks and common pitfalls: The resulting root must keep the argument positive.

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

13 / Transfer#Your turn

Solve the equation, with the domain restriction.

log⁡2(x−4)+log⁡2(x−5)=1\log_2(x-4)+\log_2(x-5)=1
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.
Hint 2
Set t=x−4; then t(t−1)=2.
Worked solution

  1. Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.

  2. Calculate or simplify this relation.

    x>5;(x−4)(x−5)=2;x=6 or 3x>5;\quad (x-4)(x-5)=2;\quad x=6\text{ or }3

  3. The smaller algebraic root makes both logarithm arguments negative.

The requested value is 6.

Checks and common pitfalls: The smaller algebraic root makes both logarithm arguments negative.

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

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

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

    Board plan

    • Choose an equivalent method and explain exclusions before calculating.
    • Positive arguments: A real logarithm requires a positive argument and a positive base unequal to one.
    • Substitution: Exponential quadratic substitution produces a positive variable; logarithm laws retain the original domain.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: A real logarithm requires a positive argument and a positive base unequal to one.
    • Expected reasoning: Exponential quadratic substitution produces a positive variable; logarithm laws retain the original domain.
    • Expected correction: A candidate produced by algebra may violate the original domain or sign condition.

    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 ↗