Knowledge Notes | Carey AP Precalculus Practice Lab

Log laws follow products and powers, with their real-domain conditions retained.

What a strong response shows

Products become sums; quotients become differences.
A power becomes a coefficient on a positive argument.
Do not split a sum inside a logarithm.

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What You Need to Know

Key idea

Log laws convert products, quotients, and powers into sums, differences, and coefficients, or reverse those moves.

Fast decision rule

Look for multiplication, division, or an exponent inside the log before choosing a law.

Worked steps

Expand a product into a sum, or compress a sum into one logarithm only after checking every argument is positive.

Common mistake

Applying a log law across addition inside an argument.

Mini check

Which operation becomes addition in a logarithm?

Learning Objectives

Key idea

2.12.A — Rewrite logarithmic expressions in equivalent forms.

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

Product becomes sum, quotient becomes difference, and a power becomes a coefficient.

Fast decision rule

Use the operation inside the logarithm to choose the corresponding law.

Worked steps

A coefficient outside a logarithm can move inside as an exponent, and an exponent inside can move outside as a coefficient.

Common mistake

Changing the base while expanding or compressing.

Mini check

What happens to a quotient inside a logarithm?

Relevant Representations

Key idea

Equivalent forms can expose multiplication, division, powers, or a single combined argument.

Fast decision rule

Choose expanded form for separate contributions and compressed form for one combined quantity.

Log laws
Inside structure	Expanded form	Compressed form
product	sum of logs	one log of a product
quotient	difference of logs	one log of a quotient
power	coefficient times log	one log with an exponent

Worked steps

Write each law beside the expression, apply one change at a time, and check the domain of each original argument.

Common mistake

Dropping an absolute-value condition or a positive-domain requirement.

Mini check

Why must original arguments stay positive?

Essential Relationships or Formulas

Key idea

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

\log_b(MN)=\log_b(M)+\log_b(N)

\log_b\!\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)

\log_b(M^p)=p\log_b(M)

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

Mark each product, quotient, and power before applying a matching log law; preserve positive arguments.

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?

Expand $\ln(8x^2)$ on the stated domain $x>0$ .

Compare your reasoning

The expansion is $\ln\left(8\right)+2 \ln\left(x\right)$ . The condition $x>0$ makes $\ln\left(x\right)$ defined; without it, the original would also allow negative $x$ and the general expansion would use $2 \ln \lvert x \rvert$ .

Use it in a real question

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

Practise this topic Check the whole unit

2.12 · Logarithmic Function Manipulation

Topic 2.12 · Logarithmic Function Manipulation

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?