Knowledge Notes | Carey AP Precalculus Practice Lab

Check the original logarithm's domain and verify the final value in the original equation.

What a strong response shows

Record logarithm restrictions before solving.
Use common bases or a valid inverse operation.
Reverse comparison when applying a decreasing exponential or logarithm.

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What You Need to Know

Key idea

Solve by rewriting into a common base, using an inverse logarithm step, or applying the same one-to-one function to both sides.

Fast decision rule

Make the unknown an exponent or a logarithm, then check domain and inequality direction.

Worked steps

Match bases when possible; otherwise take a logarithm of both sides and isolate the variable.

Common mistake

Taking a logarithm of an expression that is not positive or forgetting to reverse an inequality for a decreasing base.

Mini check

When does an exponential inequality reverse direction?

Learning Objectives

Key idea

2.13.A — Solve exponential and logarithmic equations and inequalities. 2.13.B — Construct the inverse function for exponential and 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

One-to-one behavior allows inverse operations, while domain restrictions control which solutions are valid.

Fast decision rule

For inequalities, track whether the exponential or logarithmic function is increasing or decreasing.

Worked steps

After solving, substitute the candidate into the original equation and reject any value outside a logarithm’s domain.

Common mistake

Accepting an extraneous solution created by an algebraic rewrite.

Mini check

What final check is required for a logarithmic equation?

Relevant Representations

Key idea

An equation, number-line interval, graph intersection, and inverse operation can communicate the same solution set.

Fast decision rule

Use a number line or graph after solving an inequality to verify the interval.

Equation strategy
Form	First move	Final check
same-base exponential	set exponents equal	substitute
different-base exponential	take a log	domain and approximation
log equation	rewrite or combine logs	positive arguments
inequality	track monotonicity	test intervals

Worked steps

Mark critical values, test one point in each interval, and write the solution with correct endpoint symbols.

Common mistake

Using a closed endpoint for a strict inequality or including a log argument equal to zero.

Mini check

Which graph feature can verify the number of solutions?

Essential Relationships or Formulas

Key idea

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

b^u=b^v\Rightarrow u=v\quad(b>0,\ b\ne1)

\log_b(M)=\log_b(N)\Rightarrow M=N\quad(M,N>0)

0<b<1\Rightarrow b^u\text{ is decreasing}

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

Choose common-base or logarithm methods, solve, then test domain and inequality endpoints in the original statement.

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?

Solve $\ln(x-2)=\ln(5-x)$ in the real numbers.

Compare your reasoning

The original domain is $2<x<5$ . Equal logarithms of positive arguments give $x-2=5-x$ , so $x=\frac{7}{2}$ . It lies in the original domain and both arguments equal $\frac{3}{2}$ .

Use it in a real question

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

Practise this topic Check the whole unit

2.13 · Exponential and Logarithmic Equations and Inequalities

Topic 2.13 · Exponential and Logarithmic Equations and Inequalities

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?