# Logarithm Rules

In these lessons, we will look at four basic rule of logarithms (or properties of logarithms) and how to apply them. You may want to also look at the

proofs for these properties.

Related Topics:

Proof of the Logarithm Rules,

More Algebra Lessons,

More Algebra Worksheets,

More Algebra Games,

Logarithm Games
The following table gives a summary of the logarithm rules. Scroll down the page for more explanations and examples on how to use the rules to simplify and expand logarithmic expressions.

The

rules of logarithms are

1) Product Rule

The logarithm of a product is the sum of the logarithms of the factors.

log_{a} *xy* = log_{a} *x* + log_{a} y

2) Quotient Rule

The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator

log_{a} = log_{a} x – log_{a} *y*

3) Power Rule

log_{a} *x*^{n} = *n*log_{a} x

4) Change of Base Rule

where* x *and *y* are positive, and *a* > 0, *a* ≠ 1

**Example:**

Simplify the following, expressing each as a single logarithm:

a) log _{2} 4 + log _{2} 5

b) log _{a} 28 – log _{a} 4

c) 2 log _{a} 5 – 3 log _{a} 2

**Solution:**

a) log _{2} 4 + log _{2} 5 = log _{2} (4 × 5) = log _{2} 20

b) log _{a} 28 – log _{a} 4 = log _{a} (28 ÷ 4) = log _{a} 7

c) 2 log _{a} 5 – 3 log _{a} 2 = log _{a} 5^{2} – log _{a} 2^{3} = log _{a}

*Example: *

Evaluate 2 log_{3} 5 + log_{3} 40 – 3 log_{3} 10

* Solution: *

2 log_{3} 5 + log_{3} 40 – 3 log_{3} 10

= log_{3} 5^{2} + log_{3} 40 – log_{3} 10^{3}

= log_{3} 25 + log_{3} 40 – log_{3} 1000

= log_{3}

= log_{3} 1

= 0

* Example: *

Given that log_{2} 3 = 1.585 and log_{2} 5 = 2.322, evaluate log_{4} 15

* Solution: *

## Videos

Introduction to the first two logarithm properties. Product Law & Quotient Law.

Property three and four of logarithms - Power Law & Change of Base Law.

The Properties of Logarithms

The video explains explains and applies various properties of logarithms. The main focus is how to apply the product, quotient, and power property of logarithms.

Product property: The log of a product equals the sum of the logs.

Quotient Property: The log of a quotient equals the difference of the logs.

Power Property: The log of a power equals the product of the power and the log.

Examples on how to expand logarithmic expression and how to write expressions as a single logarithm.

Basic Logarithm Properties with examples.

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.