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Unit 2 · Exponential & Logarithmic Functions Flashcards Cheat Sheet Essentials Visual Review MC Practice FRQ Practice

AP Precalculus Unit 2 Essentials

The must-know terms and core concepts for Unit 2: Exponential & Logarithmic Functions. Every vocabulary word and idea you need to master.

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Key Concept 1
Exponential change is proportional — a constant factor, not a constant amount
The heart of Unit 2 is proportional change: a quantity that grows or shrinks by the same factor over equal intervals. Arithmetic sequences and linear functions add a constant; geometric sequences and exponential functions multiply by a constant. An exponential f(x) = ab^x has an initial value a and a base b that is a growth factor (b > 1) or decay factor (0 < b < 1), and it always approaches its horizontal asymptote without reaching it.
Sequences Exponential Growth & Decay Proportional Change
Key Concept 2
The logarithm is the inverse of the exponential
A logarithm undoes an exponential: log_b(x) = y exactly when b^y = x. Because they are inverses, the graphs of y = b^x and y = log_b(x) are reflections across y = x, and their domains and ranges are swapped — the exponential's range (0, ∞) becomes the logarithm's domain. This inverse relationship, reached through composition and inverse functions, is what makes it possible to solve for an unknown exponent. The log properties (product, quotient, power, change of base) turn products into sums and exponents into coefficients.
Composition & Inverses Logarithms Log Properties
Key Concept 3
Choose a model from behavior — and validate it
Modeling is a central skill: decide whether data calls for an exponential or a logarithmic model based on how it changes, build the model (including exponential and logarithmic regressions), and then justify the choice using residuals. A semi-log plot is a powerful check — exponential data becomes a straight line when the output axis is logarithmic — and logarithmic models capture perception-based scales like pH, decibels, and earthquake magnitude.
Modeling Semi-log Plots Regression
Arithmetic sequence
A sequence with a constant common difference between consecutive terms; it behaves like a linear function.
Sequences
Geometric sequence
A sequence with a constant common ratio between consecutive terms; it behaves like an exponential function.
Sequences
Exponential function
A function of the form f(x) = ab^x, where a is the initial value and b > 0, b ≠ 1. It changes by a constant factor over equal intervals.
Exponential
Growth / decay factor
The base b of an exponential function. b > 1 gives growth; 0 < b < 1 gives decay.
Exponential
Percent rate of change
The constant percent by which an exponential changes each unit. Growth: b = 1 + r; decay: b = 1 − r.
Exponential
Horizontal asymptote
The line the graph of an exponential approaches but never reaches — y = 0 for a basic exponential.
Exponential
Composition of functions
(f ∘ g)(x) = f(g(x)): applying one function to the output of another. Its domain depends on both functions.
Composition & Inverses
Inverse function
A function that reverses the inputs and outputs of f; its graph is the reflection of f across y = x.
Composition & Inverses
One-to-one function
A function in which each output comes from exactly one input; it passes the horizontal line test and has an inverse.
Composition & Inverses
Logarithm
log_b(x) = y means b^y = x — the exponent to which base b must be raised to produce x.
Logarithms
Common logarithm
log x, the logarithm with base 10.
Logarithms
Natural logarithm
ln x, the logarithm with base e, where e ≈ 2.718.
Logarithms
Logarithmic function
y = log_b(x), the inverse of the exponential. Domain (0, ∞), range all reals, vertical asymptote x = 0.
Logarithms
Product property
log_b(MN) = log_b(M) + log_b(N).
Log Properties
Quotient property
log_b(M/N) = log_b(M) − log_b(N).
Log Properties
Power property
log_b(M^p) = p · log_b(M).
Log Properties
Change-of-base formula
log_b(x) = ln(x)/ln(b) = log(x)/log(b), used to evaluate any logarithm on a calculator.
Log Properties
Extraneous solution
A candidate solution to a logarithmic equation that must be rejected because it makes a log's argument zero or negative.
Equations
Exponential regression
A technology-based fit of an exponential model to data; residuals are examined to judge fit.
Modeling
Logarithmic regression
A technology-based fit of a logarithmic model to data.
Modeling
Semi-log plot
A plot with a logarithmic output axis; exponential data appears as a straight line.
Modeling
Residual
The difference between an observed data value and the value predicted by a model; used to validate a model choice.
Modeling