---
title: "AP Precalculus Unit 2 Review: Exponential & Log Functions"
description: "AP Precalculus Unit 2 covers Change in Linear and Exponential Functions and Exponential Functions. Study guides, practice questions, and key terms."
canonical: "https://fiveable.me/ap-pre-calc/unit-2"
type: "unit"
subject: "AP Pre-Calculus"
unit: "Unit 2 – Exponential and Logarithmic Functions"
---

# AP Precalculus Unit 2 Review: Exponential & Log Functions

## Overview

Unit 2 develops two major function families, exponential and logarithmic, starting from sequences and building through manipulation, inverses, equations, modeling, and semi-log plots. The unit accounts for 27-40% of the AP Pre-Calculus exam, making it the most heavily weighted unit in the course.

## AP CED Alignment

This unit hub is organized around AP Course and Exam Description topics, skills, and exam task types when they are available in the source data.
- 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
- 2.1: Arithmetic and Geometric Sequences
- 2.2: Linear vs. Exponential Functions
- 2.3-2.4: Exponential Function Properties and Manipulation
- 2.5-2.6: Exponential Modeling and Model Validation
- 2.9-2.10: Logarithmic Expressions and Inverses of Exponential Functions
- 2.11-2.12: Logarithmic Function Properties and Manipulation
- 2.13: Solving Exponential and Logarithmic Equations and Inequalities
- 2.14-2.15: Logarithmic Modeling and Semi-log Plots
- Practice 1 - Procedural and Symbolic Fluency
- Practice 2 - Multiple Representations
- Practice 3 - Communication and Reasoning
- FRQ 2 – Modeling a Non-Periodic Context (Calculator)
- FRQ 4 – Symbolic Manipulations (No Calculator)
- FRQ 3 – Modeling a Periodic Context (No Calculator)

## Topics

- [2.1: Change in Arithmetic and Geometric Sequences](/ap-pre-calc/unit-2/change-arithmetic-geometric-sequences/study-guide/TjmiwbtDpN420iuL): 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.
- [2.2: Change in Linear and Exponential Functions](/ap-pre-calc/unit-2/change-linear-exponential-functions/study-guide/lfvcJiKaWrIXcYHo): 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.
- [2.3: Exponential Functions](/ap-pre-calc/unit-2/exponential-functions/study-guide/5hZXVBTYwi72vxCy): 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.
- [2.4: Exponential Function Manipulation](/ap-pre-calc/unit-2/exponential-function-manipulation/study-guide/wgkA05QIw8C2351F): 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.
- [2.5: Exponential Function Context and Data Modeling](/ap-pre-calc/unit-2/exponential-function-context-data-modeling/study-guide/6gd54uXCIKjI00Sn): 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.
- [2.6: Competing Function Model Validation](/ap-pre-calc/unit-2/competing-function-model-validation/study-guide/VeTW7I04PfukXfeT): 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.
- [2.7: Composition of Functions](/ap-pre-calc/unit-2/composition-functions/study-guide/glFlt2HgsCSjvjSL): 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.
- [2.8: Inverse Functions](/ap-pre-calc/unit-2/inverse-functions/study-guide/JkTPSAR9TH5LfSXP): 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.
- [2.9: Logarithmic Expressions](/ap-pre-calc/unit-2/logarithmic-expressions/study-guide/OcOVToNeiQNxP11V): 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.
- [2.10: Inverses of Exponential Functions](/ap-pre-calc/unit-2/inverses-exponential-functions/study-guide/7mdx6zi19alJ4hK3): 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.
- [2.11: Logarithmic Functions](/ap-pre-calc/unit-2/logarithmic-functions/study-guide/U0eLmF48zLQJSMJA): 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.
- [2.12: Logarithmic Function Manipulation](/ap-pre-calc/unit-2/logarithmic-function-manipulation/study-guide/tBicQgqFwNoesetP): 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.
- [2.13: Exponential and Logarithmic Equations and Inequalities](/ap-pre-calc/unit-2/exponential-logarithmic-equations-inequalities/study-guide/Mor8iJ1w4VWPX8Wi): 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.
- [2.14: Logarithmic Function Context and Data Modeling](/ap-pre-calc/unit-2/logarithmic-function-context-data-modeling/study-guide/viEiJjFy5ba6uEyw): 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.
- [2.15: Semi-log Plots](/ap-pre-calc/unit-2/semi-log-plots/study-guide/sGsPkQ8aoU7UKBna): 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.

## Hardest Topics And Analytics

Snapshot: practice snapshot
This snapshot uses Fiveable practice activity to show where students tend to miss questions and which review moves are worth prioritizing first.
- **63% average MCQ accuracy** (Across 14k multiple-choice practice attempts for this unit.)
- **14k MCQ attempts** (Practice activity included in this snapshot.)
- **53% average FRQ score** (Across 68 scored free-response attempts for this unit.)
- **2.15: Semi-log Plots**: 46% MCQ miss rate across 601 attempts. Review Semi-log Plots with attention to how the concept appears in AP-style source and evidence questions.
- **2.14: Logarithmic Function Context and Data Modeling**: 42% MCQ miss rate across 508 attempts. Review Logarithmic Function Context and Data Modeling with attention to how the concept appears in AP-style source and evidence questions.
- **2.5: Exponential Function Context and Data Modeling**: 39% MCQ miss rate across 1489 attempts. Review Exponential Function Context and Data Modeling with attention to how the concept appears in AP-style source and evidence questions.
- **2.12: Logarithmic Function Manipulation**: 38% MCQ miss rate across 738 attempts. Review Logarithmic Function Manipulation with attention to how the concept appears in AP-style source and evidence questions.

## Review Notes

### 2.1: Arithmetic and Geometric Sequences

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.

- **Arithmetic explicit formula**: a_n = a_0 + dn, or equivalently a_n = a_k + d(n - k) when the kth term is known instead of the initial value.
- **Geometric explicit formula**: g_n = g_0 * r^n, or g_n = g_k * r^(n - k) when starting from the kth term.
- **Growth comparison**: Increasing arithmetic sequences grow by the same amount each step; increasing geometric sequences grow by a larger amount each successive step because the base is multiplied repeatedly.
- **Discrete domain**: Sequences are defined only on whole numbers, so the graph shows isolated points rather than a continuous curve.

**Checkpoint:** 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

### 2.2: Linear vs. Exponential Functions

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.

- **Linear form**: f(x) = b + mx, where b is the initial value and m is the constant rate of change (slope).
- **Exponential form**: f(x) = ab^x, where a is the initial value and b is the constant multiplicative factor per unit input.
- **Point-based exponential**: f(x) = y_i * r^(x - x_i), analogous to point-slope form for linear functions, built from a known point and ratio.
- **Two-point determination**: Both linear and exponential functions are fully determined by two distinct input-output pairs; find slope m = (y_2 - y_1)/(x_2 - x_1) or ratio b = (y_2/y_1)^(1/(x_2 - x_1)).

**Checkpoint:** 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)

### 2.3-2.4: Exponential Function Properties and Manipulation

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.

- **Horizontal asymptote**: y = 0 for f(x) = ab^x; the function approaches but never reaches zero as x goes to negative infinity (growth) or positive infinity (decay).
- **Product property graphical meaning**: b^(x+k) = b^k * b^x = ab^x, so a horizontal translation by k is equivalent to a vertical dilation by a = b^k.
- **Power property graphical meaning**: b^(cx) = (b^c)^x, so a horizontal dilation by factor 1/c is equivalent to changing the base to b^c.
- **Unit-fraction exponent**: b^(1/k) is the kth root of b, connecting rational exponents to radicals.
- **Negative exponent**: b^(-n) = 1/b^n, which converts a decay base greater than 1 to a growth base less than 1 and vice versa.

**Checkpoint:** Rewrite f(x) = 3^(x + 4) in the form ab^x and identify the vertical dilation factor.

### 2.5-2.6: Exponential Modeling and Model Validation

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.

- **Growth factor interpretation**: For f(d) = ab^d, the base b is the multiplicative factor per unit of d. Equivalent forms can reveal different time scales, such as f(d) = (b^7)^(d/7) showing the weekly factor.
- **Vertical shift to reveal proportionality**: If raw data does not show proportional outputs, adding a constant to all dependent values may reveal the exponential pattern.
- **Residual plot**: Plot of (predicted - actual) values against input; a patternless scatter indicates the model is appropriate, while a curved or systematic pattern indicates a mismatch.
- **Overestimate vs. underestimate**: Depending on context, it may be preferable for a model to consistently overestimate or underestimate; this is a contextual judgment, not just a statistical one.

**Checkpoint:** A model predicts population values and the residual plot shows a clear U-shape. What does this indicate about the chosen model?

### 2.7: Composition of Functions

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.

- **Evaluating a composition**: Compute g(x) first, then substitute that result into f. For example, if g(2) = 5 and f(5) = 11, then f(g(2)) = 11.
- **Domain restriction**: The domain of f(g(x)) includes only those x-values for which g(x) is in the domain of f.
- **Non-commutativity**: f(g(x)) and g(f(x)) are typically different functions; order matters.
- **Function decomposition**: A complex function can be split into simpler inner and outer functions; for example, h(x) = (3x + 1)^5 decomposes into f(u) = u^5 and g(x) = 3x + 1.
- **Transformation as composition**: A horizontal translation f(x + k) is the composition of f with g(x) = x + k; a horizontal dilation f(kx) is the composition of f with g(x) = kx.

**Checkpoint:** Given f(x) = 2x + 1 and g(x) = x^2, find f(g(3)) and g(f(3)) and explain why they differ.

### 2.8: Inverse Functions

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.

- **One-to-one requirement**: A function must pass the horizontal line test on its domain to have an inverse; the domain may need to be restricted.
- **Finding the inverse analytically**: Swap x and y in y = f(x), then solve for y to get y = f^(-1)(x).
- **Input-output pair reversal**: If f contains the pair (a, b), then f^(-1) contains (b, a); domain and range are exchanged.
- **Composition identity**: f(f^(-1)(x)) = x and f^(-1)(f(x)) = x on the invertible domain; this is the defining property of inverse functions.

**Checkpoint:** Find the inverse of f(x) = 3x - 7 and verify using the composition identity.

### 2.9-2.10: Logarithmic Expressions and Inverses of Exponential Functions

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.

- **Logarithm definition**: log_b(c) = a if and only if b^a = c, with b greater than 0, b not equal to 1, and c greater than 0.
- **Common and natural logarithms**: log without a base means log_10; ln means log_e where e is approximately 2.718.
- **Logarithmic scale**: Each unit on a log scale represents a multiplicative change by the base; for base 10, units correspond to 10^0, 10^1, 10^2, and so on.
- **Inverse relationship**: log_b(b^x) = x and b^(log_b(x)) = x; the two functions undo each other on their respective domains.
- **Graph reflection**: The graph of y = log_b(x) is the reflection of y = b^x across the line y = x; every point (s, t) on b^x becomes (t, s) on log_b(x).

**Checkpoint:** 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

### 2.11-2.12: Logarithmic Function Properties and Manipulation

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.

- **Product property**: log_b(xy) = log_b(x) + log_b(y); graphically, a horizontal dilation f(x) = log_b(kx) equals a vertical translation f(x) = log_b(k) + log_b(x).
- **Power property**: log_b(x^n) = n * log_b(x); raising the input to a power produces a vertical dilation of the graph.
- **Change of base formula**: log_b(x) = log_a(x) / log_a(b) for any valid base a; all logarithmic functions are vertical dilations of each other.
- **Domain and asymptote**: Domain of log_b(x) is (0, infinity); vertical asymptote at x = 0. A horizontal shift g(x) = log_b(x - h) moves the asymptote to x = h.
- **Concavity**: For a greater than 0 and b greater than 1, f(x) = a log_b(x) is concave down; for a less than 0 or 0 less than b less than 1, concavity reverses.

**Checkpoint:** Use log properties to rewrite log_2(8x^3) as a sum and product of simpler logarithmic expressions.

### 2.13: Solving Exponential and Logarithmic Equations and Inequalities

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.

- **Same-base strategy**: If both sides can be written as the same base, set the exponents equal: b^f(x) = b^g(x) implies f(x) = g(x).
- **Logarithm strategy**: When bases cannot be matched, apply ln or log to both sides and use the power property to bring the variable out of the exponent.
- **Extraneous solutions**: Solutions that emerge algebraically but make a log argument non-positive or violate domain restrictions must be discarded.
- **Inverse of transformed functions**: To find the inverse of f(x) = ab^(x-h) + k or f(x) = a log_b(x - h) + k, reverse each operation in the opposite order.
- **Inequalities**: When applying a logarithm to both sides of an inequality, the direction of the inequality depends on whether the base is greater than 1 (preserves direction) or between 0 and 1 (reverses direction).

**Checkpoint:** Solve 5^(2x - 1) = 125 and check for extraneous solutions.

### 2.14-2.15: Logarithmic Modeling and Semi-log Plots

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.

- **Logarithmic model construction**: Build f(x) = a log_b(x) from two input-output pairs by solving a system, or use logarithmic regression with technology.
- **Semi-log plot**: A graph with a logarithmic y-axis and a linear x-axis; exponential data y = ab^x appears as a straight line with slope log_n(b) and intercept log_n(a).
- **Linearization formula**: For y = ab^x, the semi-log linear model is Y = log_n(b) * x + log_n(a), where Y = log_n(y).
- **Advantage over vertical shift**: Semi-log plots reveal exponential patterns directly; no constant needs to be added to the dependent variable values before checking fit.
- **Natural logarithm in modeling**: The natural log ln(x) is especially useful for real-world models; the linearization becomes ln(y) = x * ln(b) + ln(a).

**Checkpoint:** 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.

## Study Guides

- [2.1 Change in Arithmetic and Geometric Sequences](/ap-pre-calc/unit-2/change-arithmetic-geometric-sequences/study-guide/TjmiwbtDpN420iuL)
- [2.2 Change in Linear and Exponential Functions](/ap-pre-calc/unit-2/change-linear-exponential-functions/study-guide/lfvcJiKaWrIXcYHo)
- [2.3 Exponential Functions](/ap-pre-calc/unit-2/exponential-functions/study-guide/5hZXVBTYwi72vxCy)
- [2.4 Exponential Function Manipulation](/ap-pre-calc/unit-2/exponential-function-manipulation/study-guide/wgkA05QIw8C2351F)
- [2.5 Exponential Function Context and Data Modeling](/ap-pre-calc/unit-2/exponential-function-context-data-modeling/study-guide/6gd54uXCIKjI00Sn)
- [2.6 Competing Function Model Validation](/ap-pre-calc/unit-2/competing-function-model-validation/study-guide/VeTW7I04PfukXfeT)
- [2.7 Composition of Functions](/ap-pre-calc/unit-2/composition-functions/study-guide/glFlt2HgsCSjvjSL)
- [2.8 Inverse Functions](/ap-pre-calc/unit-2/inverse-functions/study-guide/JkTPSAR9TH5LfSXP)
- [2.9 Logarithmic Expressions](/ap-pre-calc/unit-2/logarithmic-expressions/study-guide/OcOVToNeiQNxP11V)
- [2.10 Inverses of Exponential Functions](/ap-pre-calc/unit-2/inverses-exponential-functions/study-guide/7mdx6zi19alJ4hK3)
- [2.11 Logarithmic Functions](/ap-pre-calc/unit-2/logarithmic-functions/study-guide/U0eLmF48zLQJSMJA)
- [2.12 Logarithmic Function Manipulation](/ap-pre-calc/unit-2/logarithmic-function-manipulation/study-guide/tBicQgqFwNoesetP)
- [2.13 Exponential and Logarithmic Equations and Inequalities](/ap-pre-calc/unit-2/exponential-logarithmic-equations-inequalities/study-guide/Mor8iJ1w4VWPX8Wi)
- [2.14 Logarithmic Function Context and Data Modeling](/ap-pre-calc/unit-2/logarithmic-function-context-data-modeling/study-guide/viEiJjFy5ba6uEyw)
- [2.15 Semi-log Plots](/ap-pre-calc/unit-2/semi-log-plots/study-guide/sGsPkQ8aoU7UKBna)

## Practice Preview

### Multiple-choice practice

- **AP-style practice question**: Practice 1 - Procedural and Symbolic Fluency | A population of bacteria grows according to the function $$P(t) = 100(2^t)$$. A new model $$Q$$ is proposed such that $$Q(t) = P(3t)$$. Which of the following is an equivalent expression for $$Q(t)$$?
- **AP-style practice question**: Practice 2 - Multiple Representations | Given that $$\log_2(8) = 3$$, which of the following statements correctly interprets this logarithmic expression?
- **AP-style practice question**: Practice 2 - Multiple Representations | A logarithmic scale uses base 10 to represent data spanning multiple orders of magnitude. On this scale, consecutive marked units represent $$10^0, 10^1, 10^2, 10^3$$. If a value of 5,000 is plotted on this logarithmic scale, between which two consecutive marked units does it fall?
- **AP-style practice question**: Practice 2 - Multiple Representations | On a logarithmic scale with base 2, consecutive marked units represent $$2^0, 2^1, 2^2, 2^3, 2^4$$. A measurement of 20 is plotted on this scale. To determine between which two consecutive marked units this value falls, which logarithmic expression must be evaluated?
- **AP-style practice question**: Practice 2 - Multiple Representations | A sound intensity scale uses logarithms with base 10. The decibel level $$L$$ is defined by $$L = 10 \log_{10}(I)$$, where $$I$$ is the intensity ratio. If a sound has a decibel level of 30, which of the following correctly represents the intensity ratio $$I$$?
- **AP-style practice question**: Practice 3 - Communication and Reasoning | An engineer uses the model $$L(d) = 120 - 10\log_{10}(d)$$ to predict sound level L in decibels at distance d meters from a speaker. The model was constructed using measurements taken between d = 1 and d = 100 meters. Which assumption about the sound propagation mechanism is implicit in this logarithmic model, and what would happen if the assumption fails?

### FRQ practice

- **Medication concentration decay modeling**: FRQ 2 – Modeling a Non-Periodic Context (Calculator) | Medication concentration decay modeling
- **Exponential bacterial growth and logarithmic linearization**: FRQ 4 – Symbolic Manipulations (No Calculator) | Exponential bacterial growth and logarithmic linearization
- **Exponential decay of medication concentration levels**: FRQ 3 – Modeling a Periodic Context (No Calculator) | Exponential decay of medication concentration levels

## Key Terms

- **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.

## Common Mistakes

- **Confusing horizontal translation with vertical dilation in exponential functions**: 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.
- **Forgetting to check for extraneous solutions in logarithmic equations**: 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.
- **Reversing the order in function composition**: 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.
- **Misreading the base condition for growth vs. decay**: 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.
- **Applying log properties to sums inside a logarithm**: 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.

## Exam Connections

- **Interpreting function behavior from multiple representations**: 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.
- **Building and justifying models from contextual data**: 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.
- **Algebraic manipulation with exponent and logarithm properties**: 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.

## Final Review Checklist

- **Final Unit 2 review checklist**: Use this checklist to confirm you can handle every major skill in Unit 2 before exam day.
- **Write explicit formulas for arithmetic and geometric sequences**: 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).
- **Distinguish linear from exponential behavior in data**: 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.
- **Apply exponent and logarithm properties to rewrite expressions**: Use product, power, negative exponent, and change-of-base rules to convert between equivalent forms and connect algebraic rewrites to graphical transformations.
- **Find and verify inverse functions**: 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.
- **Solve exponential and logarithmic equations and check for extraneous solutions**: 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.
- **Build and validate exponential and logarithmic models from data**: Construct models from two points or regression, interpret the base as a growth factor, and use residual plots to confirm model appropriateness.
- **Interpret semi-log plots and extract model parameters**: 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.

## Study Plan

- **Step 1: Sequences and the linear-exponential connection (2.1-2.2)**: 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.
- **Step 2: Exponential function properties and manipulation (2.3-2.5)**: 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.
- **Step 3: Model validation and competing models (2.6)**: 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.
- **Step 4: Composition and inverse functions (2.7-2.8)**: 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.
- **Step 5: Logarithms, properties, equations, and modeling (2.9-2.15)**: 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.

## More Ways To Review

- [Topic study guides](/ap-pre-calc/unit-2#topics)
- [FRQ practice](/ap-pre-calc/frq-practice)
- [Cram archive videos](/cram-archives?subject=ap-pre-calculus&unit=unit-2)
- [Cheatsheets](/ap-pre-calc/cheatsheets/unit-2)
- [Key terms](/ap-pre-calc/key-terms)

## FAQs

### What topics are covered in AP Pre-Calc Unit 2?

AP 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](/ap-pre-calc/unit-2) for matched practice on every topic.

### How much of the AP Pre-Calc exam is Unit 2?

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.

### What's on the AP Pre-Calc Unit 2 progress check (MCQ and FRQ)?

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).

### How do I practice AP Pre-Calc Unit 2 FRQs?

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](/ap-pre-calc/unit-2).

### Where can I find AP Pre-Calc Unit 2 practice questions?

The best place to find AP Pre-Calc Unit 2 practice questions, including MCQ and practice test sets, is [AP Pre-Calc Unit 2](/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.

### How should I study AP Pre-Calc Unit 2?

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](/ap-pre-calc/unit-2) and do at least one timed practice set per study session.

## Structured Data

```json
{"@context":"https://schema.org","@type":"FAQPage","inLanguage":"en","mainEntity":[{"@type":"Question","@id":"https://fiveable.me/ap-pre-calc/unit-2#what-topics-are-covered-in-ap-pre-calc-unit-2","name":"What topics are covered in AP Pre-Calc Unit 2?","acceptedAnswer":{"@type":"Answer","text":"AP 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:\n- 2.1 Change in Arithmetic and Geometric Sequences\n- 2.2 Change in Linear and Exponential Functions\n- 2.3 Exponential Functions\n- 2.4 Exponential Function Manipulation\n- 2.5 Exponential Function Context and Data Modeling\n- 2.6 Competing Function Model Validation\n- 2.7 Composition of Functions\n- 2.8 Inverse Functions\n- 2.9 Logarithmic Expressions\n- 2.10 Inverses of Exponential Functions\n- 2.11 Logarithmic Functions\n- 2.12 Logarithmic Function Manipulation\n- 2.13 Exponential and Logarithmic Equations and Inequalities\n- 2.14 Logarithmic Function Context and Data Modeling\n- 2.15 Semi-log Plots See <a href=\"/ap-pre-calc/unit-2\">AP Pre-Calc Unit 2</a> for matched practice on every topic."}},{"@type":"Question","@id":"https://fiveable.me/ap-pre-calc/unit-2#how-much-of-the-ap-pre-calc-exam-is-unit-2","name":"How much of the AP Pre-Calc exam is Unit 2?","acceptedAnswer":{"@type":"Answer","text":"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."}},{"@type":"Question","@id":"https://fiveable.me/ap-pre-calc/unit-2#whats-on-the-ap-pre-calc-unit-2-progress-check-mcq-and-frq","name":"What's on the AP Pre-Calc Unit 2 progress check (MCQ and FRQ)?","acceptedAnswer":{"@type":"Answer","text":"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:\n- Exponential and Logarithmic Equations and Inequalities (2.13)\n- Logarithmic Function Context and Data Modeling (2.14)\n- Exponential Function Context and Data Modeling (2.5)\n- Composition and Inverse Functions (2.7, 2.8, 2.10)\n- Semi-log Plots (2.15) Practice with questions matched to every progress check topic at <a href=\"/ap-pre-calc/unit-2\">AP Pre-Calc Unit 2</a>."}},{"@type":"Question","@id":"https://fiveable.me/ap-pre-calc/unit-2#how-do-i-practice-ap-pre-calc-unit-2-frqs","name":"How do I practice AP Pre-Calc Unit 2 FRQs?","acceptedAnswer":{"@type":"Answer","text":"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:\n1. Write out every step, not just the answer. Graders award points for process.\n2. Practice translating word problems into exponential or logarithmic equations before solving.\n3. Review semi-log plot interpretation (2.15) separately since it's a common FRQ context.\n4. Check your work against the scoring criteria to see exactly where points are earned. Find practice FRQ sets for this unit at <a href=\"/ap-pre-calc/unit-2\">AP Pre-Calc Unit 2</a>."}},{"@type":"Question","@id":"https://fiveable.me/ap-pre-calc/unit-2#where-can-i-find-ap-pre-calc-unit-2-practice-questions","name":"Where can I find AP Pre-Calc Unit 2 practice questions?","acceptedAnswer":{"@type":"Answer","text":"The best place to find AP Pre-Calc Unit 2 practice questions, including MCQ and practice test sets, is <a href=\"/ap-pre-calc/unit-2\">AP Pre-Calc Unit 2</a>. 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."}},{"@type":"Question","@id":"https://fiveable.me/ap-pre-calc/unit-2#how-should-i-study-ap-pre-calc-unit-2","name":"How should I study AP Pre-Calc Unit 2?","acceptedAnswer":{"@type":"Answer","text":"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:\n1. Learn exponential functions (2.3-2.5) and practice graphing and transforming them before moving on.\n2. Work through composition and inverse functions (2.7, 2.8) carefully. These are the bridge to understanding logarithms.\n3. 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.\n4. Practice solving exponential and logarithmic equations and inequalities (2.13) with timed sets.\n5. 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 <a href=\"/ap-pre-calc/unit-2\">AP Pre-Calc Unit 2</a> and do at least one timed practice set per study session."}}]}
```