Study Guide

Learning Objectives

• Solve exponential and logarithmic equations and inequalities (2.13.A.1).
• Identify and reject extraneous solutions (2.13.A.2).
• Rewrite $b^x$ using another base via logs (2.13.A.3).
• Construct inverses of $ab^{x-h}+k$ and $a\log_b(x-h)+k$ (2.13.B.1–2).

Suggested Practice Sequence

1. Same-base exponential equations: $2^{x}=2^{5}$. 2. Take $\log$ or $\ln$ both sides: $3^{x}=20$. 3. Log equations: $\log_2(x-1)=3$; check Dom. 4. Inequalities: $2^{x}>8$; $\log_3 x \le 2$. 5. Extraneous hunt: $\log(x-2)+\log x=\log 3$. 6. Inverse of $f(x)=2\cdot 3^{x-1}+5$.

Classroom / Check-In Moves

• "Solve, then substitute back" ritual.
• Dom checklist before celebrating an answer.
• Undo-order cards for transformed inverses.

Misconception Watch-List

• Dropping Dom checks.
• Wrong inequality direction with decreasing logs.
• Inverse steps out of order.
• Treating $b^x=c$ as $x=c/b$.

Vocabulary to Own

• exponential equation; logarithmic equation
• inequality; extraneous solution
• change of base rewrite
• inverse function; reverse mapping
• additive transformation

Materials Map

• Practice: solve and check
• Word: contextual Dom and inequalities
• Mixed / multi-*: link to 2.12 properties and 2.8/2.10 inverses
• Review & practice test: timed solve/check fluency
• Interactive: equation and extraneous checks