Review AP Pre-Calculus Unit 2 to build fluency with exponential and logarithmic functions, from sequences and function manipulation to modeling real-world data. This unit carries 27-40% of the exam and connects directly to population growth, radioactive decay, sound intensity, and earthquake magnitude contexts.
Use the topic guides, key terms, and practice questions available for every topic in this unit to work through all 15 topics systematically.
Unit 2 spans 15 topics organized around a central idea: exponential and logarithmic functions describe situations where quantities change proportionally to their current size. The unit begins with sequences to build intuition, then moves through function properties, algebraic manipulation, inverse relationships, equation solving, data modeling, and finally semi-log plots as a tool for detecting exponential patterns in data.
Unit 2 covers exponential and logarithmic functions, their properties, inverses, algebraic manipulation, and real-world modeling. It accounts for 27-40% of the AP Pre-Calculus exam and includes topics 2.1 through 2.15.
Topics 2.1 and 2.2 establish that arithmetic sequences parallel linear functions (constant additive change) and geometric sequences parallel exponential functions (constant multiplicative change). This framing makes f(x) = ab^x feel like a natural extension of g_n = g_0 * r^n.
Topics 2.3-2.5 and 2.9-2.11 develop the two function families in parallel. Exponential functions f(x) = ab^x have domain all reals and a horizontal asymptote at y = 0. Logarithmic functions f(x) = a log_b(x) have domain (0, infinity) and a vertical asymptote at x = 0. Each is the inverse of the other.
Topics 2.5, 2.6, 2.14, and 2.15 focus on applying these functions to data. You build models from two points or regression, validate them with residual plots, and use semi-log plots to linearize exponential data without needing to add a constant first.
Every major concept in Unit 2 connects to one principle: when a quantity changes by a constant ratio over equal input intervals, an exponential model fits. When input values change proportionally over equal output intervals, a logarithmic model fits. Recognizing which direction the proportionality runs determines which function family to use and how to build, manipulate, and interpret the model.
Arithmetic sequences use a common difference d; geometric sequences use a common ratio r. Both are discrete functions on whole numbers with explicit formulas that can start from any known term.
Linear functions extend arithmetic sequences (constant additive change) and exponential functions extend geometric sequences (constant multiplicative change) to all real inputs. Both types are determined by two points.
f(x) = ab^x is monotonic, always concave in one direction, has no extrema on open intervals, no inflection points, and a horizontal asymptote at y = 0. Growth occurs when b greater than 1; decay when 0 less than b less than 1.
Product, power, negative exponent, and unit-fraction exponent rules rewrite exponential expressions. Each rule has a graphical meaning: horizontal translations become vertical dilations, and horizontal dilations become base changes.
Build f(x) = ab^x from a ratio and initial value, two input-output pairs, or exponential regression. The base b is the growth factor per unit input and connects to percent change. Equivalent forms reveal different time-scale interpretations.
Compare linear, quadratic, and exponential models using contextual clues and residual plots. A patternless residual scatter supports the model; a curved or systematic pattern indicates the wrong model type was chosen.
f(g(x)) uses g's output as f's input. Composition is not commutative. The domain is restricted to inputs of g whose outputs are in the domain of f. Transformations can be understood as compositions with g(x) = x + k or g(x) = kx.
A function is invertible on a domain where it is one-to-one. The inverse swaps input-output pairs, reflects the graph across y = x, and satisfies f(f^(-1)(x)) = x. Find it analytically by swapping x and y and solving.
log_b(c) = a means b^a = c. Common log uses base 10; natural log uses base e. Logarithmic scales represent multiplicative changes, making them useful for quantities like sound intensity and earthquake magnitude.
f(x) = log_b(x) is the inverse of g(x) = b^x. Their graphs reflect across y = x. Exponential growth means outputs multiply as inputs add; logarithmic growth means outputs add as inputs multiply.
f(x) = a log_b(x) has domain (0, infinity), range all reals, vertical asymptote at x = 0, no extrema on open intervals, and no inflection points. It is always monotonic and always concave in one direction.
Product property: log_b(xy) = log_b(x) + log_b(y). Power property: log_b(x^n) = n log_b(x). Change of base: log_b(x) = log_a(x)/log_a(b). Each property has a graphical interpretation involving dilations and translations.
Solve using exponent rules, log properties, and inverse relationships. Match bases when possible; apply ln or log otherwise. Always check for extraneous solutions caused by domain restrictions on logarithms.
Logarithmic models fit situations with proportional input growth over equal output intervals. Build from two points, a proportion and a real zero, or logarithmic regression. The natural log is especially common in real-world applications.
A semi-log plot scales the y-axis logarithmically. Exponential data appears linear, with slope log_n(b) and intercept log_n(a) for the model y = ab^x. No vertical shift of the data is needed to detect the exponential pattern.
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Review Semi-log Plots with attention to how the concept appears in AP-style source and evidence questions.
Review Logarithmic Function Context and Data Modeling with attention to how the concept appears in AP-style source and evidence questions.
Review Exponential Function Context and Data Modeling with attention to how the concept appears in AP-style source and evidence questions.
Review Logarithmic Function Manipulation with attention to how the concept appears in AP-style source and evidence questions.
A sequence is a function from the whole numbers to the real numbers, so its graph is discrete points, not a curve. Arithmetic sequences add a constant difference d each step; geometric sequences multiply by a constant ratio r each step. Both can be written from any known term, not just the initial value.
Given that a sequence has a_3 = 12 and a_7 = 28, determine whether it is arithmetic or geometric, then write its explicit formula.
| Feature | Arithmetic sequence | Geometric sequence |
|---|---|---|
| Change type | Constant difference d | Constant ratio r |
| Explicit formula | a_n = a_0 + dn | g_n = g_0 * r^n |
| Graph shape | Linear discrete points | Exponential discrete points |
| Parallel function | Linear f(x) = b + mx | Exponential f(x) = ab^x |
Linear functions extend arithmetic sequences to all real inputs; exponential functions extend geometric sequences. The key diagnostic: over equal-length input intervals, if output changes by a constant amount the function is linear; if output changes by a constant factor the function is exponential. Either type can be determined from two input-output pairs.
A data set shows outputs of 5, 10, 20, 40 at inputs 0, 1, 2, 3. Identify the function type and write its equation.
| Property | Linear f(x) = b + mx | Exponential f(x) = ab^x |
|---|---|---|
| Change over equal intervals | Constant difference | Constant ratio |
| Operation driving change | Addition | Multiplication |
| Graph shape | Straight line | Curve with asymptote |
| Determined by | Two points (slope + intercept) | Two points (ratio + initial value) |
The general form f(x) = ab^x has initial value a (a not equal to 0) and base b (b greater than 0, b not equal to 1). When a greater than 0 and b greater than 1 the function grows; when 0 less than b less than 1 it decays. Exponential functions are always monotonic, always concave in one direction, have no extrema on open intervals, and have no inflection points. Exponent rules let you rewrite expressions to reveal equivalent forms with different graphical interpretations.
Rewrite f(x) = 3^(x + 4) in the form ab^x and identify the vertical dilation factor.
Exponential models are built from a ratio and initial value, from two input-output pairs by solving a system, or from technology using exponential regression. The base b represents the growth factor per unit input and is directly tied to percent change. Competing models (linear, quadratic, exponential) are validated using residual plots: a random scatter of residuals supports the chosen model; a patterned residual plot signals a poor fit.
A model predicts population values and the residual plot shows a clear U-shape. What does this indicate about the chosen model?
The composite function (f composed with g)(x) = f(g(x)) uses the output of g as the input of f. Composition is not commutative: f(g(x)) and g(f(x)) are generally different. The domain of f composed with g is restricted to inputs of g whose outputs fall within the domain of f. Composing any function with the identity f(x) = x returns the original function unchanged.
Given f(x) = 2x + 1 and g(x) = x^2, find f(g(3)) and g(f(3)) and explain why they differ.
A function is invertible on a domain if each output maps from exactly one input (one-to-one). The inverse f^(-1) reverses the mapping: if f(a) = b then f^(-1)(b) = a. Graphically, the inverse is the reflection of f across the line y = x. The composition identity f(f^(-1)(x)) = f^(-1)(f(x)) = x confirms an inverse relationship.
Find the inverse of f(x) = 3x - 7 and verify using the composition identity.
log_b(c) = a means b^a = c: the logarithm asks what exponent on base b produces c. Logarithmic and exponential functions are inverses: f(x) = log_b(x) undoes g(x) = b^x. Their graphs are reflections across y = x. Exponential growth means outputs multiply as inputs add; logarithmic growth means outputs add as inputs multiply.
Evaluate log_3(81) without a calculator and explain using the definition of logarithm.
| Feature | Exponential g(x) = b^x | Logarithmic f(x) = log_b(x) |
|---|---|---|
| Domain | All real numbers | (0, infinity) |
| Range | (0, infinity) | All real numbers |
| Asymptote | Horizontal: y = 0 | Vertical: x = 0 |
| Growth pattern | Output multiplies as input adds | Output adds as input multiplies |
Logarithmic functions f(x) = a log_b(x) share structural properties with exponential functions: always monotonic, always concave in one direction, no extrema on open intervals, no inflection points. The three log properties (product, power, change of base) each have graphical interpretations. The natural logarithm ln(x) = log_e(x) is a special case used frequently in modeling.
Use log properties to rewrite log_2(8x^3) as a sum and product of simpler logarithmic expressions.
Solving these equations uses exponent rules, log properties, and the inverse relationship between the two function families. Always check for extraneous solutions because log arguments must be positive and exponential outputs are always positive. The identity b^x = c^((log_c(b))(x)) allows rewriting any exponential in a different base.
Solve 5^(2x - 1) = 125 and check for extraneous solutions.
Logarithmic models fit situations where input values change proportionally over equal output intervals, such as sound intensity (decibels) or earthquake magnitude (Richter scale). Models are built from two points, a proportion and a real zero, or logarithmic regression. Semi-log plots scale the y-axis logarithmically so that exponential data appears linear, making it easier to confirm an exponential model and extract its parameters without needing to add a constant to the data first.
A semi-log plot of a data set shows a straight line with slope 0.3 and intercept 1.2 (using log base 10). Write the corresponding exponential model y = ab^x.
Try AP-style multiple-choice questions and written prompts after you review the notes.
A timed, AP-style test built only from unit 2 questions — multiple choice plus free response, scored with AI feedback. Use the standard format or build your own.
| Term | Definition |
|---|---|
| sequence | A function from the whole numbers to the real numbers; its graph consists of discrete points rather than a continuous curve. |
| exponential equation | An equation in which the variable appears in an exponent, often solved by matching bases, applying logarithms, or using the inverse relationship between exponential and logarithmic functions. |
| properties of exponents | Rules for manipulating exponential expressions: product rule b^m * b^n = b^(m+n), power rule (b^m)^n = b^(mn), and negative exponent rule b^(-n) = 1/b^n. |
| product property for exponents | b^m * b^n = b^(m+n); graphically, a horizontal translation of an exponential function is equivalent to a vertical dilation. |
| power property for exponents | (b^m)^n = b^(mn); graphically, a horizontal dilation of an exponential function is equivalent to a change of base. |
| logarithm | The inverse of an exponential function; log_b(x) is the exponent to which base b must be raised to equal x, defined for b greater than 0, b not equal to 1, and x greater than 0. |
| natural logarithm | A logarithm with base e (approximately 2.718), written ln(x); the inverse of the natural exponential function e^x. |
| natural base e | The mathematical constant approximately equal to 2.718 used as the base in exponential and logarithmic models for continuous growth and decay phenomena. |
| properties of logarithms | Rules for rewriting log expressions: product rule log_b(xy) = log_b(x) + log_b(y), power rule log_b(x^n) = n log_b(x), and change of base log_b(x) = log_a(x)/log_a(b). |
| change of base formula | log_b(x) = log_a(x) / log_a(b) for any valid base a; allows conversion between bases and implies all logarithmic functions are vertical dilations of each other. |
| invertible function | A function that has an inverse; it must be one-to-one on its domain, meaning each output corresponds to exactly one input. |
| composite function | A function formed by applying one function to the output of another, written f(g(x)) or f composed with g; the domain is restricted to inputs of g whose outputs are in the domain of f. |
| extraneous solutions | Values produced algebraically that do not satisfy the original equation, typically because they make a logarithm argument non-positive or violate a contextual domain restriction. |
| residual plot | A graph of the differences between predicted and actual values plotted against input values; a patternless scatter indicates the regression model is appropriate for the data. |
| regression | A technology-based technique for fitting a function model (linear, quadratic, exponential, or logarithmic) to a data set by finding the best-fitting curve of the specified type. |
b^(x + k) does not equal b^x + b^k. The product property gives b^(x + k) = b^k * b^x, so a horizontal shift by k is a vertical dilation by b^k, not an additive shift of the output.
Algebraic steps can produce solutions that make a log argument zero or negative. Always substitute back into the original equation and discard any solution that violates the domain restriction x greater than 0.
f(g(x)) means apply g first, then f. Writing g(f(x)) gives a different function. Always identify the inner function (applied first) and the outer function (applied second) before substituting.
For f(x) = ab^x with a greater than 0: growth requires b greater than 1, not just b positive. A base between 0 and 1 produces decay. A negative value of a reflects the function across the x-axis and changes the range to negative outputs.
log_b(x + y) cannot be simplified using product or power properties. The product property applies only to log_b(xy), not to a sum inside the argument. Attempting to split log_b(x + y) is one of the most common algebraic errors in this unit.
AP Pre-Calculus tasks frequently present exponential or logarithmic functions as tables, graphs, equations, or verbal descriptions and ask you to identify characteristics such as growth vs. decay, asymptotes, domain, range, or concavity. Practice moving between all four representations for both function families.
Tasks in this unit often provide a real-world scenario (population, radioactive decay, sound intensity) and ask you to construct an exponential or logarithmic model, interpret the parameters in context, and use the model to make predictions. You may also need to justify model choice using residual plots or semi-log plots.
Rewriting expressions in equivalent forms is a recurring skill. Tasks may ask you to convert between exponential and logarithmic form, apply product or power properties to simplify or expand expressions, solve equations by applying inverses, and identify extraneous solutions. Connecting each algebraic step to its graphical meaning is a common reasoning demand.
Use this checklist to confirm you can handle every major skill in Unit 2 before exam day.
Given any two terms, identify the common difference or common ratio and write the formula in the form a_n = a_k + d(n - k) or g_n = g_k * r^(n - k).
Check whether outputs change by a constant difference (linear) or a constant ratio (exponential) over equal input intervals. Use this to select and build the correct model.
Use product, power, negative exponent, and change-of-base rules to convert between equivalent forms and connect algebraic rewrites to graphical transformations.
Swap x and y, solve for y, and confirm with the composition identity f(f^(-1)(x)) = x. Apply this to exponential and logarithmic functions specifically.
Use same-base matching or apply ln/log to both sides. After solving, verify that all solutions produce positive log arguments and are within any contextual domain.
Construct models from two points or regression, interpret the base as a growth factor, and use residual plots to confirm model appropriateness.
Recognize that a linear pattern on a semi-log plot indicates exponential data. Use the slope and intercept of the linearized form Y = log_n(b) * x + log_n(a) to recover a and b for the model y = ab^x.
Review arithmetic and geometric sequence formulas, then practice writing the corresponding linear and exponential functions from two data points. Focus on identifying constant difference vs. constant ratio in tables before moving on.
Work through the key characteristics of f(x) = ab^x (asymptote, concavity, monotonicity), then practice rewriting expressions using product, power, and negative exponent rules. Build at least two exponential models from two-point data and interpret the base as a percent change.
Practice generating residual plots with technology for linear, quadratic, and exponential fits on the same data set. Identify which model produces a patternless residual scatter and explain the choice using contextual reasoning.
Evaluate f(g(x)) and g(f(x)) for several function pairs to reinforce non-commutativity. Then practice finding inverses analytically and verifying with the composition identity. Connect this directly to the exponential-logarithm inverse relationship coming in 2.10.
Work through logarithm evaluation, the three log properties with their graphical meanings, and equation solving with extraneous solution checks. Finish by building logarithmic models from data and interpreting semi-log plots to confirm exponential fits and extract model parameters.
Open the individual guides for Unit 2 when you want a closer review of one topic.
Use AP-style practice after you review the notes so you can check what you understand.
Practice free-response reasoning and compare your answer with scoring guidance.
Watch past review streams filtered to Unit 2 when you want a video walkthrough.
Open the Fiveable one-page unit review, then explore visual cheatsheets for a quick refresher.
Estimate your broader AP score goal after you review the course and exam format.
Print the unit cheatsheets ($3) or unit exams ($5) for Unit 2. Yours to keep, no subscription needed.
get printablesAP Pre-Calc Unit 2 covers 15 topics built around exponential functions and logarithmic functions. You'll work through arithmetic and geometric sequences, exponential and linear change, exponential function manipulation and data modeling, composition of functions, inverse functions, logarithmic expressions, logarithmic function manipulation, exponential and logarithmic equations and inequalities, and semi-log plots. Here's the full topic list: - 2.1 Change in Arithmetic and Geometric Sequences - 2.2 Change in Linear and Exponential Functions - 2.3 Exponential Functions - 2.4 Exponential Function Manipulation - 2.5 Exponential Function Context and Data Modeling - 2.6 Competing Function Model Validation - 2.7 Composition of Functions - 2.8 Inverse Functions - 2.9 Logarithmic Expressions - 2.10 Inverses of Exponential Functions - 2.11 Logarithmic Functions - 2.12 Logarithmic Function Manipulation - 2.13 Exponential and Logarithmic Equations and Inequalities - 2.14 Logarithmic Function Context and Data Modeling - 2.15 Semi-log Plots See AP Pre-Calc Unit 2 for matched practice on every topic.
AP Pre-Calc Unit 2 makes up 27-40% of the AP exam, making it the heaviest-weighted unit on the test. That range covers everything from exponential functions and geometric sequences to logarithmic functions, inverse functions, and semi-log plots across 15 topics. Putting serious time into this unit pays off more than almost anything else you can do.
The AP Pre-Calc Unit 2 progress check in AP Classroom includes both MCQ and FRQ parts drawn from the unit's 15 topics. The MCQ section tests skills like identifying exponential functions, working with logarithmic expressions, and analyzing geometric sequences. The FRQ part asks you to model real-world contexts, manipulate logarithmic functions, and interpret semi-log plots or exponential equations. Topics most likely to show up on the progress check include: - Exponential and Logarithmic Equations and Inequalities (2.13) - Logarithmic Function Context and Data Modeling (2.14) - Exponential Function Context and Data Modeling (2.5) - Composition and Inverse Functions (2.7, 2.8, 2.10) - Semi-log Plots (2.15) Practice with questions matched to every progress check topic at AP Pre-Calc Unit 2.
AP Pre-Calc Unit 2 FRQs most often ask you to model a real-world situation using exponential functions or logarithmic functions, solve equations and inequalities, and interpret semi-log plots. To practice well, work through context-and-data-modeling problems from topics 2.5 and 2.14, then move to multi-step problems combining inverse functions (2.8, 2.10) with logarithmic manipulation (2.12). A solid FRQ practice routine looks like this: 1. Write out every step, not just the answer. Graders award points for process. 2. Practice translating word problems into exponential or logarithmic equations before solving. 3. Review semi-log plot interpretation (2.15) separately since it's a common FRQ context. 4. Check your work against the scoring criteria to see exactly where points are earned. Find practice FRQ sets for this unit at AP Pre-Calc Unit 2.
The best place to find AP Pre-Calc Unit 2 practice questions, including MCQ and practice test sets, is AP Pre-Calc Unit 2. That page has questions matched to all 15 topics, from geometric sequences and exponential functions in the early topics to logarithmic functions, semi-log plots, and equation solving in the later ones. For MCQ practice, focus on topics 2.3, 2.11, and 2.13, which consistently appear in multiple-choice format. For a practice test experience, work through the full topic list in order so the skills build on each other the way the real exam expects.
Start with the foundations before jumping to logarithmic functions: make sure you're solid on geometric sequences (2.1) and exponential function behavior (2.3) first, because everything in the back half of the unit builds on those ideas. Unit 2 covers 27-40% of the AP exam, so it's worth a structured plan. Here's a study approach that works: 1. Learn exponential functions (2.3-2.5) and practice graphing and transforming them before moving on. 2. Work through composition and inverse functions (2.7, 2.8) carefully. These are the bridge to understanding logarithms. 3. Study logarithmic expressions and logarithmic function manipulation (2.9, 2.11, 2.12) together. The log rules are easier when you see them as a group. 4. Practice solving exponential and logarithmic equations and inequalities (2.13) with timed sets. 5. Finish with data modeling (2.5, 2.14) and semi-log plots (2.15), which are the most common FRQ contexts. Review topic by topic at AP Pre-Calc Unit 2 and do at least one timed practice set per study session.