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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

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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}y2) Quotient Rule

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

log

= log_{a}– log_{a}x_{a}y3) Power Rule

log

_{a}x=^{n}nlog_{a}x4) Change of Base Rule

where

xandyare postive, anda> 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: *

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.

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.