Logarithm Rules

In this lesson, 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

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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.

loga xy = loga x + loga y

2) Quotient Rule

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

loga = loga x – loga y

3) Power Rule

loga xn = nloga x

4) Change of Base Rule

where x and y are postive, 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 52 – log a 23 = log a



Example:

Evaluate 2 log3 5 + log3 40 – 3 log3 10

Solution:

2 log3 5 + log3 40 – 3 log3 10
= log3 52 + log3 40 – log3 103
= log3 25 + log3 40 – log3 1000

= log3
= log3 1
= 0

Example:

Given that log2 3 = 1.585 and log2 5 = 2.322, evaluate log4 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


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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.







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