Unit 2 · Exponential and Logarithmic Functions
The Complete AP Precalculus Guide · AP PrecalculusUnit 2 · Exponential and Logarithmic Functions
- The Complete AP Precalculus Guide
- AP Precalculus
- 3 sections
Unit 2: Exponential and Logarithmic Functions accounts for 25–40% of AP Precalculus multiple-choice content. Section I has 42 multiple-choice questions in 105 minutes and contributes 62.5% of the score. For Section II's 4 free-response questions in 70 minutes (37.5%), be ready to carry the same unit skills and representations into a complete solution. Arithmetic sequences add a constant difference over equal index steps; geometric sequences multiply by a constant ratio. Exponential functions extend multiplicative change to a real input domain, while logarithms reverse the input-output action of an exponential function.
- How AP Precalculus assesses this 25–40% of the multiple-choice section · Section I: 42 MCQs in 105 min, 62.5% · Section II: 4 FRQs in 70 min, 37.5% · show the model with semi-log or ordinary graph, table of successive ratios, equation with base initial value and time scale
- Key skills Model exponential growth and decay, Solve exponential and logarithmic equations, Compare logarithmic scales and inverses
- How to study for Unit 2 This page turns semi-log or ordinary graph, table of successive ratios, equation with base initial value and time scale into one route: compare successive differences and ratios before selecting the model.
- The organizing decision distinguish additive from multiplicative change and interpret growth factors rates and logarithmic inverses
What AP Precalculus Unit 2 covers
Use this map to connect each assessed skill to the relationship or representation that makes it visible.
Model exponential growth and decay
semi-log or ordinary graph; A constant ratio over equal input increments supports an exponential modelAPPRECALC-U2-S2Solve exponential and logarithmic equations
table of successive ratios; A logarithm reverses an exponential under domain restrictionsAPPRECALC-U2-S3Compare logarithmic scales and inverses
equation with base initial value and time scale; Equivalent growth factors must be converted before rates are comparedUnit 2: Exponential and Logarithmic Functions accounts for 25–40% of AP Precalculus multiple-choice content.
Official unit name and weighting: College Board course and exam description.
Distinguish additive change from multiplicative change
Connect the published share to the unit model
Arithmetic sequences add a constant difference over equal index steps; geometric sequences multiply by a constant ratio. Exponential functions extend multiplicative change to a real input domain, while logarithms reverse the input-output action of an exponential function.
A growth or decay model must state its initial value, its positive factor, and the time represented by one exponent step. A correct factor in the wrong time unit gives a different function, even if later algebra is flawless.
When a variable appears in an exponent, logarithms can isolate it. A complete response still checks that every logarithm argument is positive and that the final input belongs to the situation's domain.
The decision that organizes this unit
Define the system and choose the route before calculating
distinguish additive from multiplicative change and interpret growth factors rates and logarithmic inverses
compare successive differences and ratios before selecting the model
Mechanism route and repair branches
Relationships to preserve
- A constant ratio over equal input increments supports an exponential model
- A logarithm reverses an exponential under domain restrictions
- Equivalent growth factors must be converted before rates are compared
Representations to read
- semi-log or ordinary graph
- table of successive ratios
- equation with base initial value and time scale
Branches to reject
- using constant difference to justify an exponential
- confusing growth factor with percent rate
- taking a logarithm of a nonpositive quantity
| Key concept | Why it's hard | What scores |
|---|---|---|
| Differences and ratios | Both can look nearly constant in a short table | Computes the relevant pattern and names the domain |
| Exponential models | The exponent counts intervals, not just the displayed time | Defines time units and writes the factor with the correct interval |
| Logarithms and inverses | Algebra can create an input outside the original domain | Shows the inverse rule and checks every restriction |
How AP Precalculus assesses Exponential and Logarithmic Functions
What a complete response must make visible
Match the task to evidence that a reader can audit, then check the most likely reasoning failure before finalizing the response.
| Task | Evidence to show | Hurdle |
|---|---|---|
| Model exponential growth and decay | semi-log or ordinary graph; A constant ratio over equal input increments supports an exponential model | using constant difference to justify an exponential |
| Solve exponential and logarithmic equations | table of successive ratios; A logarithm reverses an exponential under domain restrictions | confusing growth factor with percent rate |
| Compare logarithmic scales and inverses | equation with base initial value and time scale; Equivalent growth factors must be converted before rates are compared | taking a logarithm of a nonpositive quantity |
Resolve the Exponential and Logarithmic Functions evidence conflict
Carry the model from prompt to check
- Step 1Let t measure years and write the value as its initial amount times 1.08 raised to t divided by 3.
- Step 2Rewrite the model with an annual factor b, where b cubed equals 1.08.
- Step 3Compute b as the cube root of 1.08, approximately 1.0260.
- Step 4Interpret 1.08 as an 8 percent three-year increase and b-1 as about a 2.60 percent effective annual increase.
Key terms for Unit 2: Exponential and Logarithmic Functions
Models, uses, and boundaries
- Model Exponential Growth and Decay
- An exponential model has a constant output ratio over equal input increments Choose this formula when the prompt asks you to model exponential growth and decay and the function, domain, interval, or representation matches the symbols shown. A Model Exponential Growth and Decay solution must stop if it applies this relationship before checking its domain, interval, endpoint, sign, and stated notation conditions.
- Convert Between Exponential and Logarithmic Forms
- A logarithm is the inverse exponent under its base and domain restrictions Choose this formula when the prompt asks you to convert between exponential and logarithmic forms and the function, domain, interval, or representation matches the symbols shown. A Convert Between Exponential and Logarithmic Forms solution must stop if it applies this relationship before checking its domain, interval, endpoint, sign, and stated notation conditions.
- Translate Discrete and Continuous Growth Rates
- A per-unit growth factor b corresponds to continuous rate k equals natural log b and discrete rate r equals b minus one Choose this formula when the prompt asks you to translate discrete and continuous growth rates and the function, domain, interval, or representation matches the symbols shown. A Translate Discrete and Continuous Growth Rates solution must stop if it applies this relationship before checking its domain, interval, endpoint, sign, and stated notation conditions.
- Recognize Logistic Growth
- A logistic model approaches carrying capacity L with scale A and growth parameter k Choose this formula when the prompt asks you to recognize logistic growth and the function, domain, interval, or representation matches the symbols shown. A Recognize Logistic Growth solution must stop if it applies this relationship before checking its domain, interval, endpoint, sign, and stated notation conditions.
AP Precalculus Unit 2 FAQ
How much of AP Precalculus does Unit 2 carry?
Unit 2: Exponential and Logarithmic Functions accounts for 25–40% of AP Precalculus multiple-choice content.
What is the first move on a Exponential and Logarithmic Functions problem?
compare successive differences and ratios before selecting the model
Which relationships should I preserve?
A constant ratio over equal input increments supports an exponential model A logarithm reverses an exponential under domain restrictions Equivalent growth factors must be converted before rates are compared
Which representations should I practice?
Practice moving among semi-log or ordinary graph, table of successive ratios, equation with base initial value and time scale.
What error should I check before submitting an answer?
Check for using constant difference to justify an exponential; confusing growth factor with percent rate; taking a logarithm of a nonpositive quantity.
Evidence workshop
Continue from the free model into complete practice
The full unit guide continues with the chapter’s worked examples, figures, scoring tables, and answer checks.
- Use the index shift before applying the common difference
- Match the exponent to the length of one growth interval
- Solve an exact half-life equation by matching powers
Full unit practice. Open the complete guide for the full evidence workshop and synthesis.
Related AP Precalculus unit guides
How to study AP Precalculus Unit 2
Start with the organizing decision
Before solving, restate the decision in operational terms: distinguish additive from multiplicative change and interpret growth factors rates and logarithmic inverses. Your first written move should be to compare successive differences and ratios before selecting the model.
Practice the same idea in several representations
Rotate through semi-log or ordinary graph, table of successive ratios, equation with base initial value and time scale. Use each representation to practice Model exponential growth and decay, Solve exponential and logarithmic equations, Compare logarithmic scales and inverses, and explain what stays invariant when the surface form changes.
Turn each error into a repair check
After every attempt, audit the response for using constant difference to justify an exponential; confusing growth factor with percent rate; taking a logarithm of a nonpositive quantity. Then redo only the first step that made the reasoning diverge, keeping units, direction, and model conditions visible.
Confirm current course details in the official College Board course and exam description for the May 2027 administration.
