LEARN · EXPLAIN · REVISE

Find the domain and vertical asymptote.

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.

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

01 / Standard#Worked example

y=\log_2(x-2)

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

Use logarithmic monotonicity after enforcing a positive argument.
Calculate or simplify this relation.
$x-2>0;\quad x\to2^{+}\Rightarrow y\to-\infty$
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

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

Teacher preparation and assessment

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.

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.