Study Guide

Learning Objectives

• Apply the product property and connect $\log_b(kx)$ to a vertical shift (2.12.A.1).
• Apply the power property and connect $\log_b(x^k)$ to a vertical stretch (2.12.A.2).
• Use change of base; recognize logs as vertical dilations of each other (2.12.A.3).
• Identify $\ln x$ as $\log_e x$ (2.12.A.4).
• Expand and condense logarithmic expressions equivalently.

Suggested Practice Sequence

1. Expand $\log_b(xy)$ and $\log_b(x/y)$ (quotient via product + power). 2. Rewrite $\log_b(x^n)$ as $n\log_b x$. 3. Condense sums/differences back to single logs. 4. Change $\log_2 10$ to $\ln$ or $\log_{10}$. 5. Match $\log_b(3x)$ with $\log_b x+\log_b 3$ on a graph story.

Classroom / Check-In Moves

• Expand/condense card sort.
• Transformation matching: dilation vs translation.
• Calculator change-of-base demos.

Misconception Watch-List

• Distributing log over addition.
• $(\log x)^n$ vs $\log(x^n)$.
• Wrong change-of-base fraction orientation.
• Base of $\ln$ as 10.

Vocabulary to Own

• product property; power property; change of base
• expand; condense; equivalent form
• horizontal dilation; vertical translation; vertical dilation
• natural logarithm $\ln$; base $e$

Materials Map

• Practice: expand/condense and change base
• Word: rewrite models for interpretation
• Mixed / multi-*: link to 2.4 exp manipulation and 2.11 features
• Review & practice test: timed rewrite fluency
• Interactive: rule checks