LEARN · EXPLAIN · REVISE

Solve the logarithmic inequality with a base below one.

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

2027 JM01 考試大綱 · 8. Logarithmic and exponential functions · PDF 2 / printed page 2

TOPIC 01

Exponential and logarithmic equations

Build understanding of exponential and logarithmic equations 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_{1/2}x>3

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.

Hint 2

The base 1/2 logarithm is decreasing.

Worked solution

Enforce positive logarithm arguments and use monotonicity or a positive exponential substitution.
Calculate or simplify this relation.
$x>0;\quad x<(1/2)^{3}$
Reverse the order when converting the logarithmic comparison.

0<x<1/8.

Checks and common pitfalls: Reverse the order when converting the logarithmic comparison.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

What must be true before using the main rule for exponential and logarithmic equations?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Choose an equivalent method and explain exclusions before calculating.
Positive arguments: A real logarithm requires a positive argument and a positive base unequal to one.
Substitution: Exponential quadratic substitution produces a positive variable; logarithm laws retain the original domain.
Close with: conditions → representation → reasoning → check.

Expected reasoning: A real logarithm requires a positive argument and a positive base unequal to one.
Expected reasoning: Exponential quadratic substitution produces a positive variable; logarithm laws retain the original domain.
Expected correction: A candidate produced by algebra may violate the original domain or sign condition.