LEARN · EXPLAIN · REVISE

Disprove the claimed addition rule using positive numbers.

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_2(a+b)=\log_2a+\log_2b

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

The valid rule uses a product inside the logarithm.

Worked solution

Rewrite logarithms as exponent equations and check base and argument conditions.
Calculate or simplify this relation.
$\log_2(1+1)=1\ne0=\log_2 1+\log_2 1$
A logarithm does not distribute over addition.

Take a=b=1: the left side is 1 and the right side is 0.

Checks and common pitfalls: A logarithm does not distribute over addition.

Reasoning checklist · self / teacher assessment

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.