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.

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 xⁿ = nlog_a 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

Properties of logarithms (or Rules of logarithms) -
Professor Edward Burger explains properties of logarithms.

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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