Algebra 2 / Precalculus · Step-by-step guide

Logarithm Rules

Understand the product, quotient, and power rules for logarithms and use them to expand or condense expressions.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

A logarithm is an exponent: log⁡b(a)=c\displaystyle \log_b(a)=c means bc=a\displaystyle b^c=a. Products become sums, quotients become differences, and exponents move in front: log⁡b(Mp)=plog⁡bM\displaystyle \log_b(M^p)=p\log_bM.

log⁡b(MN)=log⁡bM+log⁡bN\displaystyle \log_b(MN)=\log_bM+\log_bN
01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

Keep the base consistent

Log rules combine logarithms only when their bases match.

2

Operations change level

Multiplication inside becomes addition outside; division becomes subtraction; a power becomes a coefficient.

3

Respect the domain

Every logarithm argument must be positive. The base must be positive and cannot equal 1.

02

See the structure

A picture makes the relationships easier to remember.

The curve passes through its initial value and rises increasingly quickly for a growth factor greater than 1.
03

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Expand

Expand log⁡2(8x3/y)\displaystyle \log_2(8x^3/y).

  1. Turn the quotient into a difference.
  2. Turn the product into a sum.
  3. Bring the exponent 3 in front.
Answerlog⁡28+3log⁡2x−log⁡2y\displaystyle \log_2 8+3\log_2x-\log_2y
Example 2Condense

Condense 2ln⁡x−ln⁡(x+1)\displaystyle 2\ln x-\ln(x+1).

  1. Move 2 up as an exponent.
  2. A difference becomes a quotient.
  3. Check that both original arguments are positive.
Answerln⁡(x2/(x+1))\displaystyle \ln(x^2/(x+1))
Example 3Evaluate

Find log⁡3(1/27)\displaystyle \log_3(1/27).

  1. Rewrite 1/27=3−3\displaystyle 1/27=3^{-3}.
  2. Ask which exponent on 3 produces the argument.
  3. Use the inverse relationship.
Answer−3
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Splitting a logarithm across addition. For example, log⁡(2+8)\displaystyle \log(2+8) is not equal to log⁡2+log⁡8\displaystyle \log 2+\log 8. The product rule works for multiplication, not addition.

  • Combining logarithms with different bases.

  • Ignoring restrictions on the arguments.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Logarithms

Practice converting forms and applying logarithm properties.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

Expand log⁡2(8x)\displaystyle \log_2(8x).

Show hint

Use the product rule.

Show worked solution
  1. log⁡2(8x)=log⁡28+log⁡2x\displaystyle \log_2(8x)=\log_2 8+\log_2 x.
  2. log⁡28=3\displaystyle \log_2 8=3.

Answer: 3+log⁡2x\displaystyle 3+\log_2x

2

Expand ln⁡(x3/y)\displaystyle \ln(x^3/y).

Show hint

Use quotient, then power.

Show worked solution
  1. ln⁡(x3/y)=ln⁡(x3)−ln⁡y\displaystyle \ln(x^3/y)=\ln(x^3)-\ln y.
  2. Move the exponent in front.

Answer: 3ln⁡x−ln⁡y\displaystyle 3\ln x-\ln y

3

Condense 2log⁡x+log⁡y\displaystyle 2\log x+\log y.

Show hint

Reverse the power and product rules.

Show worked solution
  1. 2log⁡x=log⁡(x2)\displaystyle 2\log x=\log(x^2).
  2. Add logs by multiplying arguments.

Answer: log⁡(x2y)\displaystyle \log(x^2y)

4

Condense ln⁡a−3ln⁡b\displaystyle \ln a-3\ln b.

Show hint

Move 3 to an exponent first.

Show worked solution
  1. 3ln⁡b=ln⁡(b3)\displaystyle 3\ln b=\ln(b^3).
  2. Subtract logs by dividing arguments.

Answer: ln⁡(a/b3)\displaystyle \ln(a/b^3)

5

Evaluate log⁡3(1/9)\displaystyle \log_3(1/9).

Show hint

Write 1/9 as a power of 3.

Show worked solution
  1. 1/9=3−2\displaystyle 1/9=3^{-2}.
  2. The logarithm asks for that exponent.

Answer: −2\displaystyle -2

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

A logarithm is an exponent: log⁡b(a)=c\displaystyle \log_b(a)=c means bc=a\displaystyle b^c=a. Products become sums, quotients become differences, and exponents move in front: log⁡b(Mp)=plog⁡bM\displaystyle \log_b(M^p)=p\log_bM.

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

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