Knowledge Notes | Carey AP Precalculus Practice Lab

Same-base exponentials and logarithms reverse input/output and domain/range roles. — Same-base exponentials and logarithms reverse input/output and $\frac{\text{domain}}{\text{range}}$ roles.

What a strong response shows

Keep the same valid base when inverting.
Use the positive exponential range as the logarithm domain.
Check a pair or a composition.

Loading the interactive investigation…

What You Need to Know

Key idea

The inverse of an exponential function with initial value $1$ is a logarithm with the same base.

Fast decision rule

Swap input and output, then solve for the new output using a logarithm.

Worked steps

Start with an exponential equation, exchange $x$ and $y$ , and use a logarithm to isolate $y$ .

Common mistake

Changing the base when the inverse should keep the same base.

Mini check

What function undoes raising a positive base to a power?

Learning Objectives

Key idea

2.10.A — Construct representations of the inverse of an exponential function with an initial value of $1$ .

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

Exponential and logarithmic functions are inverse operations; their graphs mirror across $y=x$ .

Fast decision rule

Keep the base, swap the roles, and state the new domain and range.

Worked steps

For an increasing exponential, the inverse is an increasing logarithm with domain $x>0$ .

Common mistake

Giving the logarithm all real inputs when its argument must be positive.

Mini check

What is the domain of a basic logarithm?

Relevant Representations

Key idea

The inverse pair can be shown with equations, a reflection graph, a table of swapped points, or a composition check.

Fast decision rule

Use two swapped points and one composition to verify the inverse.

Exponential and logarithm
Exponential	Logarithm	Role
$y=b^{x}$	$y=\log_{b}\left(x\right)$	inverse rules
domain: all real	domain: $x>0$	swap
range: $y>0$	range: all real	swap
point $\left(x,\,y\right)$	point $\left(y,\,x\right)$	reverse

Worked steps

If the exponential table contains $\left(0,\,1\right)$ and $\left(2,\,b^{2}\right)$ , the logarithm table contains $\left(1,\,0\right)$ and $\left(b^{2},\,2\right)$ .

Common mistake

Reflecting across the x-axis instead of the line $y=x$ .

Mini check

Which line is the mirror for inverse graphs?

Essential Relationships or Formulas

Key idea

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

y=b^x\iff x=\log_b(y)

f(x)=b^x\quad\Longleftrightarrow\quad f^{-1}(x)=\log_b(x)

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

Swap the variables in the exponential rule, isolate the new output with a logarithm, and state domain and range.

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?

Find the inverse of $f(x)=3\cdot2^x$ and state its domain.

Compare your reasoning

From $x=3\cdot 2^{y}$ , divide by $3$ and take log base $2$ : $f^{-1}\left(x\right)=\log_{2}\left(\frac{x}{3}\right)$ . Its domain is $x>0$ .

Use it in a real question

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

Practise this topic Check the whole unit

2.10 · Inverses of Exponential Functions

Topic 2.10 · Inverses of Exponential 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?