Carey AP PrecalculusPractice Lab
Topic study & practice

2.9 · Logarithmic Expressions

Understand the ideas before moving on. Explore distinctive knowledge notes, worked examples, and connections between representations.

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Topic 2.9 · Logarithmic Expressions

Translate logarithmic notation into an exponential statement with valid base and argument.

A logarithm returns the exponent required for its given base.

Teaching illustration · scroll to inspect
A logarithm returns the exponent required for its given base.

What a strong response shows

  • The logarithm output is the exponent.

  • The real base is positive and different from 11.

  • The real logarithm argument must be positive.

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01

What You Need to Know

Key idea

A logarithm asks for the exponent needed to create a positive number from a positive base.

Fast decision rule

Rewrite the logarithm as an exponential equation before evaluating.

Worked steps

To evaluate a logarithm, ask which exponent on the base produces the argument.

Common mistake

Taking the logarithm of a negative argument in the real number system.

Mini check

What must be true about the base and argument of a real logarithm?

02

Learning Objectives

Key idea

2.9.A — Evaluate logarithmic expressions.

Fast decision rule

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

Worked steps

11. Name the given quantities. 22. Choose the matching function or representation. 33. Show the parameter or operation that changes. 44. 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?

03

Key Concepts

Key idea

The base must be positive and not 11; the argument must be positive. Common bases include 1010 and ee.

Fast decision rule

Use powers you know first; use a calculator only after checking the domain.

Worked steps

A logarithm with a power of its own base evaluates to that exponent.

Common mistake

Treating the base and argument as interchangeable.

Mini check

What does a logarithm measure?

04

Relevant Representations

Key idea

Logarithms connect exponential equations, words about powers, and graph coordinates.

Fast decision rule

Translate log notation into an exponential statement before estimating.

Translate a logarithm
Log formExponential formQuestion answered
log⁡b(x)=y\log_{b}\left(x\right)=yby=xb^{y}=xwhich exponent?
ln⁡(x)=y\ln\left(x\right)=yey=xe^{y}=xwhich e-power?
log⁡(x)=y\log\left(x\right)=y10y=x10^{y}=xwhich 10-power?
Worked steps

Write the base, exponent, and result in one line, then use the inverse relationship to check.

Common mistake

Reading the argument as the answer instead of the exponent.

Mini check

Which part of a logarithm becomes the exponent?

05

Essential Relationships or Formulas

Key idea

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

log⁡b(x)=y  ⟺  by=x\log_b(x)=y\iff b^y=x
b>0,b≠1,x>0b>0,\quad b\ne1,\quad x>0
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?

06

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

Rewrite the logarithm as an exponential equation, identify the needed exponent, and check the base and argument.

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?

07

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?

08

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

11. Identify the pattern. 22. Write the matching form. 33. Show the key substitution or transformation. 44. 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?

Evaluate log⁡3(1/27)\log_3(1/27) without a calculator and explain the exponent.

Compare your reasoning

Since 3−3=1273^{-3}=\frac{1}{27}, log⁡3(127)=−3\log_{3}\left(\frac{1}{27}\right)=-3. The argument is positive even though the resulting logarithm is negative.

Use it in a real question

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

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