Family Overview

In Plain Language

Topic 2.9 introduces logarithms as exponents in disguise.

• Meaning: $\log_b c = a$ means $b^a = c$ (2.9.A.1).
• Restrictions: base $b>0$, $b\neq 1$; argument $c>0$ for real logs.
• Common log: if no base is written, $\log c$ means $\log_{10} c$.
• Exact vs estimate: powers of the base give exact values; other values may need technology (2.9.A.2).
• Log scale: each unit multiplies by the base (e.g., $10^0, 10^1, 10^2, \ldots$) (2.9.A.3).

Why It Matters

Logs undo exponentials (preview 2.10) and appear in pH, Richter, decibels, and scientific scales. AP items test the exponential-log equivalence, evaluation, and scale interpretation.

How Families Can Help

• Translate every log statement into an exponential: "$\log_2 8=3$ because $2^3=8$."
• Practice powers of 2, 3, 5, and 10 for exact values.
• Compare linear ticks ($0,1,2$) with log ticks ($1,10,100$).

What Not to Do

• Do not allow base $b\le 0$ or $b=1$.
• Do not evaluate $\log_b$ of a nonpositive argument in the reals.
• Do not confuse $\log_b c$ with $b/c$ or $c/b$.
• Do not treat common log as natural log unless the base is $e$.