Study Guide
Learning Objectives
- Construct representations of inverses of $y=b^x$ (initial value 1) as logarithmic functions (2.10.A.1).
- Contrast additive/multiplicative change in exponential vs logarithmic relationships (2.10.A.2).
- Verify $b^x$ and $\log_b x$ are inverses by composition (2.10.A.3).
- Relate graphs by reflection over $y=x$ (2.10.A.4).
- Swap ordered pairs between exponential and log tables/graphs (2.10.A.5).
Suggested Practice Sequence
1. Table for $g(x)=3^x$; reverse to build $\log_3$ values. 2. Compose $\log_3(3^x)$ and $3^{\log_3 x}$. 3. Reflect $y=3^x$ over $y=x$. 4. Describe: as $x$ adds 1, $3^x$ multiplies by 3; for $\log_3$, as inputs multiply by 3, outputs add 1. 5. Write $f(x)=a\log_b x$ parameters from a context or graph.
Classroom / Check-In Moves
- Pair-swap cards: $(2,9)\leftrightarrow(9,2)$ for base 3.
- "Add vs multiply" sentence frames for each family.
- Identity-composition warm-ups.
Misconception Watch-List
- Same domain for $b^x$ and $\log_b x$.
- Thinking reflection is over an axis, not $y=x$.
- Swapping growth descriptions (which multiplies / which adds).
- Writing inverse of $b^x$ as $x^b$.
Vocabulary to Own
- logarithmic function; base $b$; coefficient $a$
- inverse functions; identity composition
- reflection over $y=x$
- additive change; multiplicative change
Materials Map
- Practice: inverse pairs, composition, reflection
- Word: undo exponential models with logs
- Mixed / multi-*: link to 2.8 inverses and 2.9 log meaning
- Review & practice test: timed inverse fluency
- Interactive: pair, graph, and growth checks