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Algebra / Exponential & Logarithmic Functions

Logarithm Rules

Logarithms have three working rules: a log of a product is a sum, a log of a quotient is a difference, and an exponent inside a log comes out in front. Each is an exponent rule in disguise, which is the easiest way to remember them — and to avoid the tempting rules that are not true.

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Chapter 6 · Lesson 7 of 15Exponential & Logarithmic Functions
  1. 1.Exponential Functions
  2. 2.Graphing Exponential Functions
  3. 3.Writing Exponential Functions
  4. 4.Exponential Growth and Decay
  5. 5.Logarithms
  6. 6.Common and Natural Logarithms
  7. 7.Logarithm Rules
  8. 8.Change of Base Formula
  9. 9.Graphing Logarithmic Functions
  10. 10.Exponential vs Logarithmic Functions
  11. 11.Solving Exponential Equations
  12. 12.Solving Logarithmic Equations
  13. 13.Continuous Compound Interest
  14. 14.Exponential vs Logistic Growth
  15. 15.Richter Scale and Earthquake Intensity
See all algebra chapters →

A logarithm is an exponent, so it is no surprise that logarithms follow rules borrowed straight from the exponent rules. Those rules are what make logarithms useful: they turn multiplication into addition and powers into multiplication, which is how slide rules and log tables sped up calculation for three centuries, and how you will pull an unknown out of an exponent in the equations later in this chapter.

Product rule, quotient rule and power rule of logarithms
The product, quotient and power rules

The Rules of Logarithms

For a base , , and positive numbers and :

RuleStatement
Product rule
Quotient rule
Power rule
Log of 1
Log of the base
Inverse properties and
One-to-one property

The first three do the heavy lifting. The last four come straight from the definition of a logarithm. One more rule, for switching bases, has its own lesson: the change of base formula.

Where the Log Rules Come From: The Exponent Rules

Every log rule is an exponent rule rewritten in logarithmic form. Seeing them side by side is the best way to remember which operation turns into which:

Exponent ruleLogarithm ruleWhat it says
multiply inside add outside
divide inside subtract outside
power inside multiply outside
Each exponent rule paired with the logarithm rule it produces: product to sum, quotient to difference, power to multiple
Each log rule is an exponent rule in logarithmic form

The Product Rule: Adding Logs With the Same Base

Why it works. Let and , so and . Then , and converting back gives .

Read right to left, the product rule says adding logs with the same base gives the log of the product: . That is the basis of log multiplication — the product of two numbers can be found by adding their logarithms.

The Quotient Rule: Dividing Inside a Logarithm

The quotient rule handles division inside the log. Be careful with the phrase “dividing logarithms”: — one log divided by another — is not . A quotient of two logs is a change of base, not a quotient rule: .

The Power Rule

The log power rule moves an exponent inside the logarithm out to the front as a multiplier. It is the single most important rule for solving equations, because it brings an unknown exponent down to where you can reach it: .

It also handles roots, since a root is a fractional power: .

The One-to-One Property

Because a logarithmic function is one-to-one, two logs with the same base can only be equal if their arguments are equal. This lets you drop the logs from both sides of an equation such as , leaving .

Example 1: Using the Log Rules to Evaluate

ExpressionRule AppliedResult
product:
product:
quotient:
power:
power:
product:

Notice the last row: neither nor is a nice number on its own, but their sum is exactly . Combining first is often the only way to evaluate an expression by hand.

Expanding Logarithms

Expanding means using the rules to break one logarithm into a sum and difference of simpler logs. Work from the outside in: quotients first, then products, then powers.

ExpressionExpanded Form

The last row keeps intact — a sum inside a log cannot be split.

Condensing Logarithms

Condensing runs the rules in reverse to combine several logs into one. Apply the power rule first (coefficients become exponents), then combine sums into products and differences into quotients.

ExpressionCondensed Form

Condensing is what you do before solving a logarithmic equation, so that each side is a single log you can undo. Expanding and condensing are also called simplifying logarithms.

ln of x cubed y over z squared expands to 3 ln x plus ln y minus 2 ln z, and condensing reverses it
Expanding and condensing are the same rules run in opposite directions

Worked Example

Expand as far as possible.

Step 1 — apply the quotient rule to split the fraction:

Step 2 — apply the product rule to the numerator:

Step 3 — rewrite the root as a power:

Step 4 — apply the power rule to both powers:

Step 5 — evaluate the numerical log, since :

Common Mistakes to Avoid

  • Splitting a sum. . The product rule is for .
  • Turning a quotient of logs into a difference. . The quotient rule is about , with the division inside.
  • Applying the power rule to a power of the log. . The power rule needs the exponent on the argument: .
  • Combining logs with different bases. cannot be merged with the product rule. Convert to a common base first.

Practice Problems

Work each one before opening the answer.

Problem 1. Evaluate .

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Step 1 — combine with the product rule:

Step 2 — :

Answer:

Problem 2. Evaluate .

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Step 1 — combine with the quotient rule:

Step 2 — :

Answer:

Problem 3. Expand .

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Step 1 — product rule:

Step 2 — power rule, and :

Answer:

Problem 4. Expand .

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Step 1 — quotient rule:

Step 2 — product rule on the denominator, keeping the subtraction on both terms:

Step 3 — power rule:

Answer:

Problem 5. Condense into a single logarithm.

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Step 1 — power rule, coefficients become exponents:

Step 2 — sums become products, differences become quotients:

Answer:

Problem 6. Given and , find and .

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Step 1 — write and expand:

Step 2 — write and use the quotient rule:

Answer: ,

The one rule not covered here converts a logarithm from one base to another — essential when your calculator only has and . That is the change of base formula.

Frequently Asked Questions

Can you split the log of a sum, like log(x + y)?+

No. There is no rule for — the product rule applies to , not to a sum. For example, , while . A sum inside a logarithm has to stay as it is.

How are the log rules related to the exponent rules?+

Each log rule is an exponent rule read backwards. Multiplying powers adds exponents, , so the log of a product is the sum of the logs. Dividing subtracts exponents, giving the quotient rule, and gives the power rule. A logarithm is an exponent, so it follows exponent arithmetic.

Do the logarithm rules work for ln and log?+

Yes. The rules hold for every valid base, so and exactly as with any other base. The only requirement is that every term uses the same base — you cannot combine with the product rule.

Related lessons

LogarithmsExponent RulesCommon and Natural LogarithmsChange of Base Formula