Knowledge Notes | Carey AP Precalculus Practice Lab

For a real logarithm, solve positivity of the whole argument before evaluating.

What a strong response shows

Require the complete argument to be positive.
For a standard shifted log, its argument-zero boundary is the vertical asymptote.
The base and any reflection determine direction.

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What You Need to Know

Key idea

A logarithmic function is the inverse of an exponential function. Its input must be positive, and its graph has a vertical asymptote.

Fast decision rule

Read the base first: base above $1$ gives increasing behavior; base between $0$ and $1$ gives decreasing behavior.

Worked steps

For a basic logarithm, the input approaches zero from the right while the output changes without bound.

Common mistake

Drawing a logarithm on both sides of the vertical asymptote.

Mini check

Why is the logarithm’s domain restricted?

Learning Objectives

Key idea

2.11.A — Identify key characteristics of logarithmic functions.

Fast decision rule

Match the prompt verb to the representation it asks you to build or interpret.

Worked steps

$1$ . Name the given quantities. $2$ . Choose the matching function or representation. $3$ . Show the parameter or operation that changes. $4$ . Check one value in context.

Common mistake

Starting calculations before identifying what the input and output mean.

Mini check

Can you state what the input, output, and requested evidence are before choosing a formula?

Key Concepts

Key idea

Important features are domain, range, vertical asymptote, intercept, monotonicity, and transformations.

Fast decision rule

Find the argument equal to zero to locate the vertical asymptote after transformations.

Worked steps

For a shifted form, set the inside expression equal to zero, then apply the outside shift to the range or output.

Common mistake

Using the exponential horizontal asymptote rule for a logarithm.

Mini check

What type of asymptote does a logarithm have?

Relevant Representations

Key idea

The formula, graph, table, and inverse exponential story should agree about domain and direction.

Fast decision rule

Locate the vertical asymptote before reading intercepts or end behavior.

Logarithmic characteristics
Feature	Basic form	Transformed form
Domain	$x>0$	inside argument>0
Range	all real	all real after vertical shifts
Asymptote	$x=0$	inside argument=0
Direction	base>1 increases	base between $0$ and $1$ decreases

Worked steps

Sketch the asymptote, choose a point on the valid side, and check whether outputs increase or decrease.

Common mistake

Plotting points on the forbidden side of the asymptote.

Mini check

Which feature tells you where the graph cannot cross?

Essential Relationships or Formulas

Key idea

Use an exact relationship to expose the pattern before substituting values.

f(x)=\log_b(x),\qquad b>0,\ b\ne1

x>0,\qquad -\infty<y<\infty

Fast decision rule

Keep the variables and restrictions visible, then substitute with units attached.

Worked example

Write the relationship, identify each parameter, substitute the known values, and check one output.

Common mistake

Using a familiar formula without checking its domain, base, interval, or unit assumptions.

Mini check

Which symbol or parameter carries the feature the prompt asks you to find?

Worked Examples

Key idea

A short, labeled chain of steps is easier to check than a page of unexplained algebra.

Fast decision rule

Write the representation first, calculate second, and interpret last.

Worked steps

Find the vertical asymptote from the logarithm argument, then use one valid point to identify direction and placement.

Common mistake

Skipping the interpretation step and reporting an answer with no units, domain, or model evidence.

Mini check

Which step would you show so another student could reproduce the result?

Common Mistakes

Key idea

Most errors come from a mismatch between the pattern, the operation, and the allowed input.

Fast decision rule

Before submitting, check pattern, operation, domain, and units in that order.

Worked steps

Circle the first line where the pattern or restriction changes. Correct that line, then recompute only the affected steps.

Common mistake

Trusting a calculator output that hides a wrong model, sign, base, or domain.

Mini check

What independent check would catch your most likely error?

AP Exam Tips

Key idea

Use the topic workflow to connect the representation, the equation, and the context before calculating.

Fast decision rule

Read the prompt, choose the smallest reliable representation, and verify the result in context.

Worked steps

$1$ . Identify the pattern. $2$ . Write the matching form. $3$ . Show the key substitution or transformation. $4$ . Check units, domain, and direction.

Common mistake

Submitting a number without explaining why the selected model or transformation fits.

Mini check

Can you name the evidence that supports your final model or answer?

Practice

Open this topic’s practice set to apply the method. Work independently, then compare the evidence in your reasoning with the checked explanation.

TRY IT IN A NEW SETTING

Can you transfer the method?

For $y=\log_2(x-1)$ , state the domain, vertical asymptote and x-intercept.

Compare your reasoning

The domain is $x>1$ , and the vertical asymptote is $x=1$ . Setting the output to zero gives $x-1=1$ , so the x-intercept is $\left(2,\,0\right)$ .

Use it in a real question

Work independently, then compare the mathematical evidence in your answer.

Practise this topic Check the whole unit

2.11 · Logarithmic Functions

Topic 2.11 · Logarithmic Functions

What a strong response shows

What You Need to Know

Learning Objectives

Key Concepts

Relevant Representations

Essential Relationships or Formulas

Worked Examples

Common Mistakes

AP Exam Tips

Can you transfer the method?