Family Overview

In Plain Language

Topic 2.11 is about the key characteristics of logarithmic functions $f(x)=a\log_b x$.

• Domain / range: Dom $=(0,\infty)$; Range $=\mathbb{R}$ (2.11.A.1).
• Shape: always increasing or always decreasing; always concave one way — no global extrema on $\mathbb{R}$ and no inflection points (2.11.A.2).
• Proportional inputs: if an additive shift $g(x)=f(x+k)$ still shows proportional inputs over equal output intervals, then $f$ is logarithmic (2.11.A.3).
• Asymptote / ends: vertical asymptote $x=0$; end behavior unbounded as $x\to 0^+$ and as $x\to\infty$ (2.11.A.4).

Why It Matters

After logs as expressions (2.9) and as inverses of exponentials (2.10), students must describe log graphs the way they described $ab^x$ in 2.3. AP items ask for domain, asymptotes, monotonicity, concavity, and end-behavior limits.

How Families Can Help

• Checklist: Dom, Range, VA $x=0$, increasing/decreasing, concave up/down.
• Sketch $y=\log_2 x$ and $y=\log_{1/2} x$.
• Ask: "Is there a turning point on the whole domain?" (No.)

What Not to Do

• Do not give Dom as all reals.
• Do not claim a vertex or inflection point on $(0,\infty)$.
• Do not forget the vertical asymptote at $x=0$.
• Do not confuse horizontal asymptotes (exponential) with the log's vertical asymptote.