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

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 / 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.

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 ↗