Study Guide

Learning Objectives

• State Dom and Range of general-form logarithmic functions (2.11.A.1).
• Describe monotonicity and concavity; reject false extrema/inflection claims (2.11.A.2).
• Use proportional-input tests related to additive transformations (2.11.A.3).
• Identify VA $x=0$ and unbounded end behavior with limits (2.11.A.4).

Suggested Practice Sequence

1. Dom/Range for $a\log_b x$. 2. Classify increasing ($b>1$, $a>0$) vs decreasing cases. 3. State "no extrema / no inflection on Dom." 4. Write $\lim_{x\to 0^+}\log_b x$ and $\lim_{x\to\infty}\log_b x$ for $b>1$ and $0<b<1$. 5. Sketch with VA $x=0$ and point $(1,0)$.

Classroom / Check-In Moves

• Characteristic card sort vs exponential cards.
• Limit sentence frames for $x\to 0^+$ and $x\to\infty$.
• Closed-interval extrema only as a special case.

Misconception Watch-List

• Dom includes 0 or negatives.
• Horizontal asymptote language copied from exponentials.
• Claiming an inflection because the curve bends.
• Thinking every log is increasing.

Vocabulary to Own

• logarithmic function; general form $a\log_b x$
• domain; range; vertical asymptote
• monotonic; concave up/down
• end behavior; limit; extrema; inflection

Materials Map

• Practice: characteristic identification
• Word: contexts with Dom $>0$ and asymptote meaning
• Mixed / multi-*: link to 2.3 / 2.10 inverses
• Review & practice test: timed feature fluency
• Interactive: quick characteristic checks