AP Pre-Calculus Unit 2 ReviewExponential and Logarithmic Functions

AP Pre-Calc Unit 2 is about functions that change by repeated multiplication instead of repeated addition. The single biggest idea is proportional change, meaning output values grow or shrink by the same factor over equal input intervals, which is exactly what exponential functions model and what logarithmic functions reverse. The unit covers sequences, exponential and logarithmic functions, composition and inverses, and data modeling with semi-log plots. At 27-40% of the AP exam, it carries more weight than any other unit in the course.

What this unit covers

From sequences to functions: two kinds of change

The unit opens by contrasting the two fundamental growth patterns.

• An arithmetic sequence adds the same amount each step. Its general term is $a_n = a_0 + dn$, and it behaves like a linear function $f(x) = b + mx$ with discrete inputs.
• A geometric sequence multiplies by the same ratio each step. Its general term is $g_n = g_0 r^n$, and it behaves like an exponential function $f(x) = ab^x$ with discrete inputs.
• The test for which one you have is always the same. Over equal-length input intervals, constant differences in output mean linear, constant ratios mean exponential. This single check shows up constantly in table-based problems.
• Both sequence types can be written from a known kth term instead of the initial term, using $a_n = a_k + d(n-k)$ or $g_n = g_k r^{(n-k)}$, which mirrors point-slope form for lines.

Exponential functions and their algebra

• The general form is $f(x) = ab^x$ with initial value $a \neq 0$ and base $b > 0$, $b \neq 1$. Growth happens when $a > 0$ and $b > 1$; decay happens when $a > 0$ and $0 < b < 1$.
• Exponential graphs are always increasing or always decreasing, always concave up or always concave down, with no extrema and no inflection points. End behavior involves a horizontal asymptote.
• Exponent properties have graphical meaning, and the exam loves this. The product property $b^m b^n = b^{m+n}$ means a horizontal translation of an exponential function is the same as a vertical dilation. The power property $(b^m)^n = b^{mn}$ means a horizontal dilation is the same as changing the base.
• Equivalent forms reveal different information. If $f(d) = 2^d$ models doubling every day, then $f(d) = (2^7)^{d/7} = 128^{d/7}$ shows the quantity multiplies by 128 every week. Same function, different story.

Composition and inverses, the bridge to logarithms

This is the conceptual hinge of the unit. Logarithms only make sense once you understand inverses.

• The composite function $(f \circ g)(x) = f(g(x))$ feeds the outputs of $g$ into $f$. Its domain is restricted to inputs of $g$ whose outputs land in the domain of $f$.
• You also decompose functions, rewriting a complicated function as a composition of simpler ones. Translations and dilations are themselves compositions with $g(x) = x + k$ or $g(x) = kx$.
• A function is invertible on a domain where each output comes from exactly one input. The inverse reverses the mapping, so if $f(a) = b$ then $f^{-1}(b) = a$, tables flip their columns, and graphs reflect over $y = x$.
• The defining property is $f(f^{-1}(x)) = f^{-1}(f(x)) = x$, and the domain and range swap between a function and its inverse.

Logarithms: definition, properties, solving

• The definition is everything. $\log_b c = a$ if and only if $b^a = c$. A logarithm is just an exponent in disguise. Base 10 is the common log (written $\log x$), and base $e$ is the natural log.
• Logarithmic functions $f(x) = a\log_b x$ have domain $x > 0$, range all reals, a vertical asymptote at $x = 0$, and are always increasing or always decreasing with consistent concavity, the mirror image of exponential behavior.
• The log properties parallel the exponent properties. Product: $\log_b(xy) = \log_b x + \log_b y$. Power: $\log_b x^n = n\log_b x$. Change of base: $\log_b x = \frac{\log_a x}{\log_a b}$. Each one has a graphical interpretation too (for example, a horizontal dilation of a log graph is a vertical translation).
• Solving equations means using the inverse relationship to free up the variable, then checking for extraneous solutions. A log equation can produce an answer that makes the argument of a log negative, and that answer must be thrown out.
• You also build inverses of transformed functions. The inverse of $f(x) = ab^{x-h} + k$ comes from undoing each operation in reverse order, and the same goes for $f(x) = a\log_b(x - h) + k$.

Modeling with data: choosing and validating

• Exponential models fit situations with proportional growth, like populations, compound interest, and radioactive decay. You can build one from an initial value and a ratio, or from two input-output pairs. Sometimes a constant must be added to the dependent variable to reveal the proportional pattern.
• Logarithmic models fit the reverse situation, where inputs change proportionally over equal output intervals. Earthquake magnitude and sound intensity are the classic contexts.
• When linear, quadratic, and exponential models all seem plausible, you compare them using context and residuals. A model is justified when its residual plot shows no pattern. The error is predicted minus actual, and context decides whether overestimating or underestimating is worse.
• Semi-log plots are the diagnostic tool. Put the y-axis on a log scale, and exponential data turns linear. For $y = ab^x$, the linearized model is $y = (\log_n b)x + \log_n a$, so the slope of the semi-log line is $\log_n b$ and the intercept is $\log_n a$.

Unit 2, Exponential and Logarithmic Functions at a glance

Why Unit 2, Exponential and Logarithmic Functions matters in AP Pre-Calc

The whole course is organized around how functions change, and Unit 2 supplies the second great pattern of change. Unit 1 gave you functions built on rates of change that themselves change additively. This unit gives you proportional change, and then hands you the logarithm as the tool that undoes it.

• Composition and inverses are introduced here but belong to the entire course. Every inverse function you meet later, including inverse trig functions, runs on the machinery built in this unit.
• Modeling is one of the course's three big mathematical practices, and this unit is where model selection and validation (residual plots, error analysis, semi-log linearization) get their fullest treatment.
• The "equivalent forms reveal different properties" theme appears here in its sharpest form. Rewriting $2^d$ as $128^{d/7}$ or expanding a log expression is the same skill the exam rewards everywhere.

How this unit connects across the course

• Backward to polynomial and rational functions (Unit 1): the rate-of-change language carries straight over. Unit 1 functions change by varying amounts; exponential functions change by a constant factor. Competing model validation explicitly pits linear and quadratic models from Unit 1 against exponential models from this unit.
• Forward to trigonometric and polar functions (Unit 3): inverse trig functions (arcsin, arccos, arctan) require restricting a domain to make a function invertible, the exact idea from inverse functions here. If inverses click now, Unit 3's hardest concept gets much easier.
• Forward to parameters, vectors, and matrices (Unit 4): function composition returns when you work with transformations and with functions involving parameters, and the modeling-validation mindset carries into every later modeling task.

Key formulas and procedures

• $a_n = a_0 + dn$ and $a_n = a_k + d(n-k)$: arithmetic sequence from an initial value or from any known term.
• $g_n = g_0 r^n$ and $g_n = g_k r^{(n-k)}$: geometric sequence from an initial value or from any known term.
• $f(x) = ab^x$: general exponential form, where $a$ is the initial value and $b$ is the growth factor tied to percent change.
• $b^m b^n = b^{m+n}$ and $(b^m)^n = b^{mn}$: exponent properties for rewriting exponentials, each with a graphical meaning (translation as dilation, dilation as base change).
• $\log_b c = a \iff b^a = c$: the definition of a logarithm; convert between forms whenever a variable is stuck in an exponent or inside a log.
• $\log_b(xy) = \log_b x + \log_b y$, $\log_b x^n = n\log_b x$: product and power properties for condensing or expanding log expressions.
• $\log_b x = \frac{\log_a x}{\log_a b}$: change of base, used to evaluate logs in any base on a calculator.
• $f(f^{-1}(x)) = x$: the identity-composition check that two functions are inverses.
• Finding an inverse of $f(x) = ab^{x-h} + k$ or $f(x) = a\log_b(x - h) + k$: undo each operation in reverse order.
• Building an exponential model from two points $(x_1, y_1)$ and $(x_2, y_2)$: solve for $b$ from the ratio of outputs, then back out $a$.
• Semi-log linearization of $y = ab^x$: the line $y = (\log_n b)x + \log_n a$, so slope gives $\log_n b$ and intercept gives $\log_n a$.
• Residual plot check: a model fits when residuals show no pattern.

Unit 2, Exponential and Logarithmic Functions on the AP exam

This unit is 27-40% of the exam, the largest share of any unit, so fluency here moves your score more than anything else. Exponential and logarithmic content appears in both multiple choice sections (with and without calculator) and in the free response section.

• Function-concept questions ask you to identify growth versus decay, describe end behavior and concavity, reason about asymptotes, and interpret what the base $b$ means in context (a growth factor or percent change).
• Manipulation questions ask you to rewrite exponential and log expressions in equivalent forms, connect those rewrites to graph transformations, and solve equations while ruling out extraneous solutions.
• Modeling free response is where this unit pays off. The exam's modeling task gives you data or a contextual scenario and asks you to construct a function model, use it to answer questions, and interpret quantities like the growth factor or error in context. Semi-log plots and residual analysis fit naturally here.
• Multiple representations matter. Expect to move between tables, graphs, equations, and verbal descriptions, for example reading inverse values off a table or estimating a composite function value from two graphs.
• On the calculator-allowed sections, regression and numerical evaluation are fair game, but you still have to justify why a chosen model is appropriate.

Essential questions

• What distinguishes a quantity that changes proportionally from one that changes at a constant rate, and how can you tell from a table, graph, or context?
• Why does every exponential relationship have a logarithmic counterpart, and what does it mean for two functions to undo each other?
• How do equivalent algebraic forms of the same function reveal different information about a real situation?
• How do you decide which function model best fits a data set, and how do you know when a model should be trusted?

Key terms to know

• Common difference: the constant amount added between successive terms of an arithmetic sequence.
• Common ratio: the constant factor multiplied between successive terms of a geometric sequence.
• Growth factor: the base $b$ in $f(x) = ab^x$, telling you what the output is multiplied by for each unit increase in input.
• Exponential decay: the pattern produced when $a > 0$ and $0 < b < 1$, so outputs shrink by a constant proportion.
• Composite function: the function $f \circ g$ that uses outputs of $g$ as inputs to $f$, with domain restricted accordingly.
• Invertible: describes a function whose every output comes from exactly one input on a given domain, so an inverse function exists.
• Logarithm: the exponent to which a base must be raised to produce a given value; $\log_b c = a$ exactly when $b^a = c$.
• Common logarithm: a logarithm with base 10, written without a subscript.
• Natural logarithm: a logarithm with base $e$, written $\ln x$.
• Extraneous solution: an answer produced by valid algebra that fails the original equation, often by making a log argument nonpositive.
• Residual plot: the graph of differences between actual and predicted values; a patternless plot justifies the model.
• Semi-log plot: a graph with one logarithmically scaled axis, on which exponential data appears linear.
• Linearization: rewriting exponential data as a linear relationship on a semi-log scale so linear techniques apply.

Common mix-ups

• A logarithm's domain restriction applies to the argument, not the answer. $\log_b x$ needs $x > 0$, but the output can be any real number, including negatives. Students flip this constantly.
• Exponential functions $ab^x$ have a variable exponent. Power functions like $x^2$ have a variable base. "It has an exponent" does not make a function exponential.
• $\log_b(x + y)$ is not $\log_b x + \log_b y$. The product property only splits multiplication inside the log, never addition.
• The inverse of $f$ is $f^{-1}$, a reversed mapping, not $\frac{1}{f(x)}$. The notation looks like a reciprocal but means something completely different.