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United StatesPrecalculus

Unit 2: Exponential and Logarithmic Functions

15 dot points across 15 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do arithmetic and geometric sequences differ in the way their terms change?

Topic 2.1 Change in Arithmetic and Geometric Sequences: define arithmetic sequences by a constant common difference and geometric sequences by a constant common ratio, and write their explicit and recursive forms.
A focused answer to AP Precalculus Topic 2.1, covering arithmetic sequences with a common difference, geometric sequences with a common ratio, and their explicit and recursive formulas.
9 min answer →

What is the fundamental difference between linear change and exponential change?

Topic 2.2 Change in Linear and Exponential Functions: contrast linear functions, which change by a constant amount, with exponential functions, which change by a constant percentage or factor over equal-length input intervals.
A focused answer to AP Precalculus Topic 2.2, covering the constant-difference behavior of linear functions versus the constant-ratio behavior of exponential functions, and how to tell them apart from data.
9 min answer →

When two models both fit data, how do you decide which is better using residuals?

Topic 2.6 Competing Function Model Validation: compare competing function models for a data set by analyzing residuals, and validate or reject a model based on the pattern and size of its residuals.
A focused answer to AP Precalculus Topic 2.6, covering residuals, residual plots, how a random residual pattern validates a model, and how to choose between competing linear, quadratic and exponential fits.
9 min answer →

How do you combine two functions by composition, and what is the domain of the result?

Topic 2.7 Composition of Functions: form the composition of two functions, evaluate it, decompose a composite function, and determine the domain of a composition.
A focused answer to AP Precalculus Topic 2.7, covering how to form and evaluate the composition of two functions, decompose a composite into inner and outer functions, and find the domain of a composition.
9 min answer →

How do you solve equations and inequalities that have the variable in an exponent or inside a logarithm?

Topic 2.13 Exponential and Logarithmic Equations and Inequalities: solve exponential and logarithmic equations and inequalities using inverse operations, the logarithm properties, and checks for extraneous solutions.
A focused answer to AP Precalculus Topic 2.13, covering solving exponential equations by taking logs, solving logarithmic equations by exponentiating, checking for extraneous solutions, and handling inequalities.
10 min answer →

How do you build an exponential model from a context or data, and interpret its parameters?

Topic 2.5 Exponential Function Context and Data Modeling: construct an exponential model from a context or data set, interpret the initial value and growth or decay factor, and use the model to make predictions.
A focused answer to AP Precalculus Topic 2.5, covering how to build an exponential model from two points or a context, interpret the initial value and growth factor, and use exponential regression to fit data.
9 min answer →

How do the exponent rules let you rewrite an exponential expression into a more useful equivalent form?

Topic 2.4 Exponential Function Manipulation: rewrite exponential expressions using the product, power, negative-exponent and rational-exponent properties to reveal equivalent forms.
A focused answer to AP Precalculus Topic 2.4, covering the product, quotient, power, negative-exponent and rational-exponent rules, and how rewriting an exponential reveals a different base, growth rate or initial value.
9 min answer →

What are the defining features of an exponential function, and how do its base and initial value shape its graph?

Topic 2.3 Exponential Functions: define exponential functions, describe how the base and initial value determine growth or decay, and analyze the domain, range and horizontal asymptote of the graph.
A focused answer to AP Precalculus Topic 2.3, covering the form of an exponential function, growth versus decay, the horizontal asymptote, domain and range, and the natural base e.
9 min answer →

What is the inverse of a function, and how do you find and verify it?

Topic 2.8 Inverse Functions: determine whether a function has an inverse, find the inverse by swapping input and output, and verify an inverse using composition and the reflection over the line y = x.
A focused answer to AP Precalculus Topic 2.8, covering one-to-one functions and the horizontal line test, finding an inverse by swapping variables, verifying with composition, and the reflection over y = x.
9 min answer →

Why is the logarithm the inverse of the exponential, and what does that reveal about its graph?

Topic 2.10 Inverses of Exponential Functions: construct the inverse of an exponential function as a logarithmic function, and relate the graph, domain and range of each to the other.
A focused answer to AP Precalculus Topic 2.10, covering how the logarithm is the inverse of the exponential, finding the inverse by swapping variables, and how their graphs reflect over y = x with swapped domain and range.
8 min answer →

What does a logarithm mean, and how do you evaluate logarithmic expressions?

Topic 2.9 Logarithmic Expressions: define a logarithm as the exponent that produces a given value, and evaluate logarithmic expressions by rewriting them in exponential form.
A focused answer to AP Precalculus Topic 2.9, covering the definition of a logarithm as an exponent, converting between logarithmic and exponential form, common and natural logs, and evaluating logarithmic expressions.
8 min answer →

When is a logarithmic model appropriate, and how do you build and interpret one?

Topic 2.14 Logarithmic Function Context and Data Modeling: construct a logarithmic model from a context or data set, interpret its parameters, and use it to make predictions.
A focused answer to AP Precalculus Topic 2.14, covering when a logarithmic model fits, building a model from a context or by logarithmic regression, interpreting its parameters, and applications such as pH and decibels.
9 min answer →

How do the logarithm properties let you expand or condense a logarithmic expression?

Topic 2.12 Logarithmic Function Manipulation: rewrite logarithmic expressions using the product, quotient, power and change-of-base properties to expand or condense them.
A focused answer to AP Precalculus Topic 2.12, covering the product, quotient, power and change-of-base properties of logarithms, and how to expand a single log or condense several into one.
9 min answer →

What are the defining features of a logarithmic function and its graph?

Topic 2.11 Logarithmic Functions: analyze the parent logarithmic function and its transformations, including its domain, range, vertical asymptote, and increasing or decreasing behavior.
A focused answer to AP Precalculus Topic 2.11, covering the parent logarithmic function, its domain and range, the vertical asymptote, growth versus the base, and transformations of logarithmic graphs.
9 min answer →

How does plotting data on a semi-log scale reveal whether it is exponential, and how do you read such a plot?

Topic 2.15 Semi-log Plots: use a semi-log plot to determine whether an exponential model is appropriate, and interpret the slope and intercept of the resulting line.
A focused answer to AP Precalculus Topic 2.15, covering how a semi-log plot linearises exponential data, why exponential data appear as a line, and how the slope and intercept relate to the base and initial value.
9 min answer →