Family Overview

In Plain Language

Topic 2.13 is about solving exponential and logarithmic equations and inequalities, checking solutions, and building inverses of transformed models.

• Solve: use exponent/log properties and the inverse relationship $b^x \leftrightarrow \log_b$ (2.13.A.1).
• Check: reject extraneous solutions that break Dom or context (2.13.A.2).
• Rewrite exponentials: $b^x = c^{x\log_c b}$ can reveal useful structure (2.13.A.3).
• Inverses of transforms: undo $f(x)=ab^{x-h}+k$ and $f(x)=a\log_b(x-h)+k$ by reversing operations (2.13.B.1–2).

Why It Matters

This is where Unit 2 skills become equation-solving power for modeling and FRQs. AP items mix algebra, Dom checks, and inverse construction for shifted/stretched models.

How Families Can Help

• Same base or take log both sides — then check Dom.
• For inequalities, test a point or use mono behavior.
• Practice: reverse a transformation chain step by step.

What Not to Do

• Do not keep solutions that make log arguments nonpositive.
• Do not forget to reverse inequality sense only when multiplying/dividing by a negative (or when applying a decreasing log/exp carefully).
• Do not invent inverses without undoing $+k$, $\cdot a$, and shifts in order.