Family Overview
In Plain Language
Topic 2.12 is about rewriting logarithmic expressions into equivalent forms—and linking those rules to graph transformations.
- Product property: $\log_b(xy)=\log_b x+\log_b y$. So $\log_b(kx)=\log_b k+\log_b x$: a horizontal dilation becomes a vertical translation (2.12.A.1).
- Power property: $\log_b(x^n)=n\log_b x$. Raising the input to a power becomes a vertical dilation (2.12.A.2).
- Change of base: $\log_b x=\dfrac{\log_a x}{\log_a b}$. All log graphs are vertical dilations of each other (2.12.A.3).
- Natural log: $\ln x=\log_e x$ (2.12.A.4).
Why It Matters
These rewrites prepare equation solving (2.13) and let students move between calculator-friendly bases and graph forms. AP items test symbolic manipulation and the transformation meanings.
How Families Can Help
- Drill: expand products; condense sums; pull exponents out front.
- Graph link: $\log(kx)$ vs $\log x+\log k$; $\log(x^k)$ vs $k\log x$.
- Practice change of base for $\log_2$ on a base-10 calculator.
What Not to Do
- Do not write $\log_b(x+y)=\log_b x+\log_b y$.
- Do not confuse $\log_b(x^n)$ with $(\log_b x)^n$.
- Do not forget arguments must stay positive.
- Do not treat $\ln$ as base 10.