LEARN · EXPLAIN · REVISE

Evaluate the logarithm with a base below one.

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

高一必修第一册(A版).pdf · 4.3 · PDF 129 / printed page 122

TOPIC 01

Logarithms

Build understanding of logarithms 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#Your turn

\log_{1/3}9

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Rewrite logarithms as exponent equations and check base and argument conditions.

Hint 2

A negative exponent turns the small base into a large value.

Worked solution

Rewrite logarithms as exponent equations and check base and argument conditions.
Calculate or simplify this relation.
$(1/3)^{-2}=9$
Positive arguments can have negative logarithms.

The requested value is -2.

Checks and common pitfalls: Positive arguments can have negative logarithms.

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

Teacher preparation and assessment

Interpret and use logarithms with domain checks.
Inverse definition: A logarithm is the exponent needed to produce a positive argument.
$\log_a b=c\iff a^c=b$
Laws: Products become sums; quotients become differences; use only positive arguments.
Close with: conditions → representation → reasoning → check.

Expected reasoning: A logarithm is the exponent needed to produce a positive argument.
Expected reasoning: Products become sums; quotients become differences; use only positive arguments.
Expected correction: The logarithm of a sum does not split into a sum of logarithms.