Logarithm rules
In this lesson, we will look at four basic rule of logarithms (or properties of logarithms) and how to apply them.
The rules of logarithms are
1) Product Law
The logarithm of a product is the sum of the logarithms of the factors.
loga xy = loga x + loga y
2) Quotient Law
The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator
loga = loga x – loga y
3) Power Law
loga xn = nloga x
4) Change of Base Law

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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