LEARN · EXPLAIN · REVISE

Find the base-10 logarithmic increase when a positive quantity is multiplied by 1000.

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

高一必修第一册(A版).pdf · 4.5 · PDF 149 / printed page 142

TOPIC 01

Applications of functions II

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

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

01 / Foundation#Your turn

\log_{10}(1000x)-\log_{10}x

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate the context into an exponential or logarithmic equation and check what the model assumes.

Hint 2

A ratio becomes a logarithmic difference.

Worked solution

Translate the context into an exponential or logarithmic equation and check what the model assumes.
Calculate or simplify this relation.
$\log_{10}(1000x/x)=\log_{10}1000=3$
The result is independent of the initial positive x.

The requested value is 3.

Checks and common pitfalls: The result is independent of the initial positive x.

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

Teacher preparation and assessment

Interpret and use applications of functions ii with domain checks.
Multiplicative change: Constant ratios lead to exponential models; logarithms solve elapsed-time questions.
Zeros and evidence: Continuity plus a sign change gives a zero; model validity still needs evidence.
Close with: conditions → representation → reasoning → check.

Expected reasoning: Constant ratios lead to exponential models; logarithms solve elapsed-time questions.
Expected reasoning: Continuity plus a sign change gives a zero; model validity still needs evidence.
Expected correction: A fitted trend does not justify unlimited extrapolation.