Kenneth ChengMathematical modeling
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Compare the values when 0<x<y.

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

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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.

Try it. Leave your reasoning visible.

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

01 / 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.

Focus on one question

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.

No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

Curriculum and source notes ↗