What it covers: Proportional change — exponential functions (growth and decay) and their inverses, logarithmic functions, plus composition, inverses, and modeling.
Exam weight: About 27–40% of the AP Precalculus exam.
The big question: How do exponential and logarithmic functions — inverses of one another — model quantities that change by a constant factor?
Arithmetic sequences add a common difference (linear). Geometric sequences multiply by a common ratio (exponential). Linear = equal differences; exponential = equal ratios over equal intervals.
Exponential Functions
f(x) = ab^x, with a = initial value and base b > 0, b ≠ 1. Domain: all reals; range: (0, ∞); horizontal asymptote y = 0.
Growth vs. Decay
b > 1 → growth; 0 < b < 1 → decay. For percent rate r: growth b = 1 + r, decay b = 1 − r.
Composition & Inverses
(f ∘ g)(x) = f(g(x)). An inverse swaps inputs and outputs and reflects the graph over y = x; only one-to-one functions (horizontal line test) have inverses.
Logarithms
log_b(x) = y ⇔ b^y = x. The log is the inverse of the exponential. Domain (0, ∞), range all reals, vertical asymptote x = 0. ln = base e, log = base 10.
Log Properties
Product: log(MN) = log M + log N. Quotient: log(M/N) = log M − log N. Power: log(M^p) = p·log M. Change of base: log_b(x) = ln x / ln b.
Solving Equations
Exponential: take a log of both sides → x = log_b(c). Logarithmic: rewrite in exponential form or combine logs, then check for extraneous solutions (arguments must be > 0).
Modeling & Semi-log
Fit exponential/logarithmic models and validate with residuals. On a semi-log plot (log output axis) exponential data becomes a straight line.
The key terms you must know
Common difference / common ratio — the constant added (arithmetic) or multiplied (geometric) between consecutive terms.
Exponential function — f(x) = ab^x; changes by a constant factor per unit input.
Growth / decay factor — the base b; b > 1 grows, 0 < b < 1 decays.
Horizontal asymptote — y = 0 for a basic exponential (shifts with a vertical translation).
Composition — (f ∘ g)(x) = f(g(x)); its domain depends on both functions.
Inverse function — reverses inputs and outputs; reflects over y = x; requires one-to-one.
Logarithm — log_b(x) = y means b^y = x; the inverse of the exponential.
Natural / common log — ln = base e; log = base 10.
Log properties — product (sum), quotient (difference), power (coefficient), change of base.
Extraneous solution — a candidate solution rejected because it makes a log's argument ≤ 0.
Semi-log plot — a plot with a logarithmic output axis; linearizes exponential data.
Key themes to remember
Exponential and logarithm are inverses. Everything about logs comes from undoing exponentials — reflect over y = x, swap domain and range.
Constant ratio, not constant difference. Exponential change multiplies; that single idea drives growth, decay, and interest.
Log properties turn products into sums. They are the tools that make exponential and logarithmic equations solvable.
Model, then justify. Choose exponential or logarithmic based on the data's behavior and defend the choice with residuals.
Represent four ways. Sequences, tables, graphs, and equations of exponentials should all be connected.
Common exam traps
Growth vs. decay base. b > 1 grows; 0 < b < 1 decays. A base like 0.85 is decay (a 15% decrease), not growth.
Inside changes reflect, don't shift, for inverses. The inverse reflects over y = x — it is not a translation.
log_b of a non-positive number is undefined. Always check that every argument stays positive; reject extraneous solutions.
The power property needs the exponent on the whole argument. log(M^p) = p·log M, but log(M) + log(N) ≠ log(M+N).
Equal ratios signal exponential, equal differences signal linear. Read the table before choosing a model.
Semi-log linearity means exponential. A straight line on a semi-log plot indicates exponential data, not linear data.
ln and log are different bases. ln is base e; log (with no base written) is base 10. Use change of base to switch.