Family Overview

In Plain Language

Topic 2.10 connects exponentials and logarithms as inverse functions.

• Log form: $f(x)=a\log_b x$ with $b>0$, $b\neq 1$, and $a\neq 0$ (2.10.A.1).
• Inverse pair: $g(x)=b^x$ and $f(x)=\log_b x$ undo each other: $f(g(x))=g(f(x))=x$ (2.10.A.3).
• Growth pattern flip: exponential — outputs multiply as inputs add; logarithmic — outputs add as inputs multiply (2.10.A.2).
• Graph: $y=\log_b x$ is the reflection of $y=b^x$ over $y=x$ (2.10.A.4).
• Ordered pairs: if $(s,t)$ is on $y=b^x$, then $(t,s)$ is on $y=\log_b x$ (2.10.A.5).

Why It Matters

This is the bridge from exponential modeling (2.3–2.5) and inverses (2.8) into full logarithmic functions (2.11). AP items test inverse recognition, pair swapping, reflection, and the additive/multiplicative growth contrast.

How Families Can Help

• Build a table for $2^x$, then reverse pairs for $\log_2 x$.
• Sketch $y=2^x$ and its reflection over $y=x$.
• Check: $\log_2(2^5)=5$ and $2^{\log_2 8}=8$.

What Not to Do

• Do not confuse $a\log_b x$ with $\log_b(ax)$ without log rules (those come in 2.12).
• Do not claim $b^x$ and $\log_b x$ share the same domain.
• Do not forget Dom$(\log_b x)=(0,\infty)$.
• Do not mix up which variable changes multiplicatively.