Kenneth ChengMathematical modeling
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Logarithmic functions

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

高一必修 第一册(A版).pdf · 4.4 · PDF 137 / printed page 130

Revisit first: Logarithms

TOPIC 01

Logarithmic functions

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

What you will be able to explain

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

Domain and inverse

Logarithmic input is positive; output ranges over the reals.

Monotonicity

The base decides whether comparisons preserve or reverse order.

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 logarithmic functions 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

Compare the values when 0<x<y.

f(t)=log⁡2tf(t)=\log_2t
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use logarithmic monotonicity after enforcing a positive argument.
Hint 2
A base greater than one gives an increasing logarithm.
Worked solution

  1. Use logarithmic monotonicity after enforcing a positive argument.

  2. Calculate or simplify this relation.

    2>1;x<y⇒log⁡2x<log⁡2y2>1;\quad x<y\Rightarrow\log_2x<\log_2y

  3. Both inputs must be positive.

f(x)<f(y).

Checks and common pitfalls: Both inputs must be 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.

02 / Standard#Worked example

Find the domain and vertical asymptote.

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

Working and explanation

BUILD THE REASONING

Hint 1
Use logarithmic monotonicity after enforcing a positive argument.
Hint 2
Locate where the argument approaches zero from above.
Worked solution

  1. Use logarithmic monotonicity after enforcing a positive argument.

  2. Calculate or simplify this relation.

    x−2>0;x→2+⇒y→−∞x-2>0;\quad x\to2^{+}\Rightarrow y\to-\infty

  3. The boundary line is not part of the graph.

Domain x>2; asymptote x=2.

Checks and common pitfalls: The boundary line is not part of the graph.

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.

03 / Transfer#Worked example

Explain why the graphs y=log₂x and y=2ˣ are reflections in y=x.

y=log⁡2x  ⟺  x=2yy=\log_2x\iff x=2^y
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use logarithmic monotonicity after enforcing a positive argument.
Hint 2
Use the same point written in both relations.
Worked solution

  1. Use logarithmic monotonicity after enforcing a positive argument.

  2. Calculate or simplify this relation.

    (a,b)↦(b,a)(a,b)\mapsto(b,a)

  3. The axes and domains are exchanged, not simply translated.

The input-output coordinates are exchanged by inversion.

Checks and common pitfalls: The axes and domains are exchanged, not simply translated.

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

Compare the values when 0<x<y.

f(t)=log⁡3tf(t)=\log_3t
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use logarithmic monotonicity after enforcing a positive argument.
Hint 2
A base greater than one gives an increasing logarithm.
Worked solution

  1. Use logarithmic monotonicity after enforcing a positive argument.

  2. Calculate or simplify this relation.

    3>1;x<y⇒log⁡3x<log⁡3y3>1;\quad x<y\Rightarrow\log_3x<\log_3y

  3. Both inputs must be positive.

f(x)<f(y).

Checks and common pitfalls: Both inputs must be 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.

05 / Foundation#Your turn

Find the domain and vertical asymptote.

y=log⁡2(x−3)y=\log_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
Use logarithmic monotonicity after enforcing a positive argument.
Hint 2
Locate where the argument approaches zero from above.
Worked solution

  1. Use logarithmic monotonicity after enforcing a positive argument.

  2. Calculate or simplify this relation.

    x−3>0;x→3+⇒y→−∞x-3>0;\quad x\to3^{+}\Rightarrow y\to-\infty

  3. The boundary line is not part of the graph.

Domain x>3; asymptote x=3.

Checks and common pitfalls: The boundary line is not part of the graph.

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.

06 / Foundation#Your turn

Find the inverse function and its domain.

y=log⁡3xy=\log_3x
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use logarithmic monotonicity after enforcing a positive argument.
Hint 2
Swap input and output after solving for the original input.
Worked solution

  1. Use logarithmic monotonicity after enforcing a positive argument.

  2. Calculate or simplify this relation.

    x=3y⇒f−1(x)=3xx=3^y\Rightarrow f^{-1}(x)=3^x

  3. The original logarithm range becomes the inverse domain.

Inverse: y=3^x for all real x.

Checks and common pitfalls: The original logarithm range becomes the inverse 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.

07 / Foundation#Your turn

Solve the inequality.

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

Working and explanation

BUILD THE REASONING

Hint 1
Use logarithmic monotonicity after enforcing a positive argument.
Hint 2
The logarithm decreases, so the comparison reverses.
Worked solution

  1. Use logarithmic monotonicity after enforcing a positive argument.

  2. Calculate or simplify this relation.

    0<x−3<1/20<x-3<1/2

  3. The argument positivity supplies the lower bound.

3<x<3+1/2.

Checks and common pitfalls: The argument positivity supplies the lower bound.

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

Find the x-intercept.

y=log⁡2(x−3)y=\log_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
Use logarithmic monotonicity after enforcing a positive argument.
Hint 2
A logarithm is zero exactly when its argument is one.
Worked solution

  1. Use logarithmic monotonicity after enforcing a positive argument.

  2. Calculate or simplify this relation.

    0=log⁡2(x−3)⇒x−3=1⇒x=40=\log_2(x-3)\Rightarrow x-3=1\Rightarrow x=4

  3. The intercept must lie inside the domain.

The requested value is 4.

Checks and common pitfalls: The intercept must lie inside the domain.

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

09 / Standard#Your turn

Find all m for which the expression is defined for every x in [0,1].

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

Working and explanation

BUILD THE REASONING

Hint 1
Use logarithmic monotonicity after enforcing a positive argument.
Hint 2
Check the smallest possible argument.
Worked solution

  1. Use logarithmic monotonicity after enforcing a positive argument.

  2. Calculate or simplify this relation.

    min⁡0≤x≤1(x+m)=m>0\min_{0\le x\le1}(x+m)=m>0

  3. Checking only an interior x can miss a forbidden endpoint.

m>0.

Checks and common pitfalls: Checking only an interior x can miss a forbidden endpoint.

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.

10 / Standard#Your turn

State the conditions under which these logarithms are reciprocals, then prove it.

log⁡ax⋅log⁡xa=1\log_a x\cdot\log_x a=1
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use one common logarithm base.
Hint 2
Both denominators must be nonzero.
Worked solution

  1. Use one common logarithm base.

  2. Calculate or simplify this relation.

    ln⁡xln⁡aln⁡aln⁡x=1\frac{\ln x}{\ln a}\frac{\ln a}{\ln x}=1

  3. x=1 is allowed as an argument of logₐx but not as the base of logₓa.

a>0, a≠1, x>0, x≠1.

Checks and common pitfalls: x=1 is allowed as an argument of logₐx but not as the base of logₓa.

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

Compare the values when 0<x<y.

f(t)=log⁡4tf(t)=\log_4t
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use logarithmic monotonicity after enforcing a positive argument.
Hint 2
A base greater than one gives an increasing logarithm.
Worked solution

  1. Use logarithmic monotonicity after enforcing a positive argument.

  2. Calculate or simplify this relation.

    4>1;x<y⇒log⁡4x<log⁡4y4>1;\quad x<y\Rightarrow\log_4x<\log_4y

  3. Both inputs must be positive.

f(x)<f(y).

Checks and common pitfalls: Both inputs must be 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.

12 / Transfer#Your turn

Find the domain and vertical asymptote.

y=log⁡2(x−4)y=\log_2(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
Use logarithmic monotonicity after enforcing a positive argument.
Hint 2
Locate where the argument approaches zero from above.
Worked solution

  1. Use logarithmic monotonicity after enforcing a positive argument.

  2. Calculate or simplify this relation.

    x−4>0;x→4+⇒y→−∞x-4>0;\quad x\to4^{+}\Rightarrow y\to-\infty

  3. The boundary line is not part of the graph.

Domain x>4; asymptote x=4.

Checks and common pitfalls: The boundary line is not part of the graph.

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.

13 / Transfer#Your turn

Find the inverse function and its domain.

y=log⁡4xy=\log_4x
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use logarithmic monotonicity after enforcing a positive argument.
Hint 2
Swap input and output after solving for the original input.
Worked solution

  1. Use logarithmic monotonicity after enforcing a positive argument.

  2. Calculate or simplify this relation.

    x=4y⇒f−1(x)=4xx=4^y\Rightarrow f^{-1}(x)=4^x

  3. The original logarithm range becomes the inverse domain.

Inverse: y=4^x for all real x.

Checks and common pitfalls: The original logarithm range becomes the inverse 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.

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 logarithmic functions?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Interpret and use logarithmic functions with domain checks.
    • Domain and inverse: Logarithmic input is positive; output ranges over the reals.
    • Monotonicity: The base decides whether comparisons preserve or reverse order.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Logarithmic input is positive; output ranges over the reals.
    • Expected reasoning: The base decides whether comparisons preserve or reverse order.
    • Expected correction: Never forget argument positivity when solving inequalities.

    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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