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Properties of Logs

Author: Sophia
PDF Version

what's covered
In this lesson, you will learn how to convert a logarithmic expression using the properties of logs. Specifically, this lesson will cover:

Table of Contents

1. Product Property of Logs

When we learned about exponents, we learned that there are certain properties with exponents that can be applied when exponential expressions are multiplied, divided, and raised to exponent powers. There are similar properties with logarithms.

The Product Property of Logarithms allows us to split a logarithmic expression into several logs which are added together. More specifically, we break down the argument of the log (what is inside the parentheses) into factors. The logarithm is then applied to each individual factor, and the string of logs are added together. Below is the general property:

formula to know
Product Property of Logs
log subscript b left parenthesis x y right parenthesis space equals space log subscript b left parenthesis x right parenthesis plus log subscript b left parenthesis y right parenthesis

We can use this property to break a quantity down into more than two factors.

EXAMPLE

Expand the logarithmic expression log subscript 4 open parentheses 90 close parentheses.

log subscript 4 open parentheses 90 close parentheses Rewrite 90 with prime factorization
log subscript 4 open parentheses 2 times 3 times 3 times 5 close parentheses Apply the Product Property of Logs
log subscript 4 open parentheses 2 close parentheses plus log subscript 4 open parentheses 3 close parentheses plus log subscript 4 open parentheses 3 close parentheses plus log subscript 4 open parentheses 5 close parentheses Combine like terms
log subscript 4 open parentheses 2 close parentheses plus 2 log subscript 4 open parentheses 3 close parentheses plus log subscript 4 open parentheses 5 close parentheses Our solution

We can also use the Product Property in the other direction to simplify logarithmic expressions. Be sure that all of the logarithms have the same base, otherwise, we cannot use the property:

EXAMPLE

Simplify the logarithmic expression log subscript 5 open parentheses 2 close parentheses plus log subscript 5 open parentheses 3 close parentheses plus log subscript 5 open parentheses 7 close parentheses.

log subscript 5 open parentheses 2 close parentheses plus log subscript 5 open parentheses 3 close parentheses plus log subscript 5 open parentheses 7 close parentheses Apply the Product Property of Logs
log subscript 5 open parentheses 2 times 3 times 7 close parentheses Evaluate parentheses
log subscript 5 open parentheses 42 close parentheses Our solution

2. Quotient Property of Logs

A property very similar to the product property exists with quotients. If we recognize a quotient within a logarithm, we can create individual logs connected with subtraction. The main difference here between the product and quotient properties is that the product property of logs connects individual logs with addition, and the quotient property of logs connects individual logs with subtraction.

formula to know
Quotient Property of Logs
log subscript b left parenthesis x over y right parenthesis equals log subscript b left parenthesis x right parenthesis space minus space log subscript b left parenthesis y right parenthesis

This property can be useful if there is a quotient explicitly written within a logarithm, especially if you can recognize a log that can be easily evaluated mentally:

EXAMPLE

Expand the logarithmic expression log subscript 4 open parentheses 16 over 3 close parentheses.

log subscript 4 open parentheses 16 over 3 close parentheses Apply the Quotient Property of Logs
log subscript 4 open parentheses 16 close parentheses minus log subscript 4 open parentheses 3 close parentheses Simplify log subscript 4 open parentheses 16 close parentheses equals 2 because 4 squared equals 16
2 minus log subscript 4 open parentheses 3 close parentheses Our solution

We can also apply this property in the other direction:

EXAMPLE

Simplify the logarithmic expression log subscript 2 open parentheses 72 close parentheses minus log subscript 2 open parentheses 9 close parentheses.

log subscript 2 open parentheses 72 close parentheses minus log subscript 2 open parentheses 9 close parentheses Apply the Quotient Property of Logs
log subscript 2 open parentheses 72 over 9 close parentheses Evaluate parentheses
log subscript 2 open parentheses 8 close parentheses Simplify log subscript 2 open parentheses 8 close parentheses equals 3 because 2 cubed equals 8
3 Our solution

3. Power Property of Logs

The power property of logs is useful when simplifying logarithmic expressions that contain an exponent within the operation. This property allows us to move the exponent outside of the log operation, and place it as a scalar multiplier to the log. In general, we write the power property as:

formula to know
Power Product of Logs
log subscript b left parenthesis x to the power of n right parenthesis equals n times log subscript b left parenthesis x right parenthesis

EXAMPLE

Evaluate the logarithmic expression log subscript 2 open parentheses 16 close parentheses.

log subscript 2 open parentheses 16 close parentheses Rewrite 16 as 2 to the power of 4
log subscript 2 open parentheses 2 to the power of 4 close parentheses Apply the Power Property of Logs
4 log subscript 2 open parentheses 2 close parentheses Rewrite log subscript 2 open parentheses 2 close parentheses as 1
4 Our solution

hint
As we saw in this example, if the argument of the log and the base are the same (both were 2 in the example above), the logarithm evaluates to 1. This relationship is explained later in this lesson.

4. Change of Base Property

The change of base property is extremely useful in using calculators to evaluate logarithmic expressions. Most calculators that can evaluate logs have only two buttons: a common log button, which is a base 10 log, and a natural log button, which is a base e log. e is a mathematical constant approximately equal to 2.718282.

What do we do, then, when we want to evaluate a log with base 3 using our calculator? Or a log with base 7? Or any other log with a base other than 10 or e? We use the change of base formula:

formula to know
Change of Base Property of Logs
log subscript b left parenthesis x right parenthesis space equals space fraction numerator log subscript a left parenthesis x right parenthesis over denominator log subscript a left parenthesis b right parenthesis end fraction

Basically, we can take the common log or natural log of the argument, no matter what the base is, but then we must divide that expression by the common log or natural log (whichever was used for the numerator) by the original log's base. Here is a concrete example:

EXAMPLE

Rewrite the logarithmic expression log subscript 3 open parentheses 40 close parentheses as a common log.

log subscript 3 open parentheses 40 close parentheses Apply Change of Base Property of Logs
fraction numerator log open parentheses 40 close parentheses over denominator log open parentheses 3 close parentheses end fraction Our solution

As we can see, log subscript 3 open parentheses 40 close parentheses on its own is difficult to evaluate using a calculator. However, thanks to the Change of Base formula, we can enter log open parentheses 40 close parentheses divided by log open parentheses 3 close parentheses into the calculator to evaluate the expression. This is helpful when given a problem like log subscript 2 open parentheses 2 close parentheses plus log subscript 2 open parentheses 4 close parentheses plus log subscript 2 open parentheses 16 close parentheses. You can rewrite this and then use the log function on the calculator:

EXAMPLE

Calculate the logarithmic expression log subscript 2 open parentheses 2 close parentheses plus log subscript 2 open parentheses 4 close parentheses plus log subscript 2 open parentheses 16 close parentheses.

log subscript 2 open parentheses 2 close parentheses plus log subscript 2 open parentheses 4 close parentheses plus log subscript 2 open parentheses 16 close parentheses Apply Change of Base Property of Logs
fraction numerator log open parentheses 2 close parentheses over denominator log open parentheses 2 close parentheses end fraction plus fraction numerator log begin display style open parentheses 4 close parentheses end style over denominator log begin display style open parentheses 2 close parentheses end style end fraction plus fraction numerator log begin display style open parentheses 16 close parentheses end style over denominator log begin display style open parentheses 2 close parentheses end style end fraction Evaluate each log with a calculator
1 plus 2 plus 4 Add
7 Our solution

5. Other Logarithmic Properties

Finally, let's review two other properties of logs that can help us simplify expressions.

formula to know
Other Properties of Logs
table attributes columnalign left end attributes row cell log subscript b left parenthesis b right parenthesis equals 1 end cell row cell log subscript b left parenthesis 1 right parenthesis equals 0 end cell end table

The first states that if the argument of the log and the base are the same, the expression evaluates to 1.

EXAMPLE

log subscript 5 open parentheses 5 close parentheses equals 1

The last property states that the log of 1, no matter what the base is, evaluates to zero.

EXAMPLE

log subscript 3 open parentheses 1 close parentheses equals 0 or log subscript 7 open parentheses 1 close parentheses equals 0

summary
There are many properties of logs that are useful when simplifying and solving logarithmic expressions and equations. The product property of logs allows us to rewrite the expression using addition, while the quotient property of logs allows us to rewrite the expression using subtraction. The power property of logs states that if there is an exponent within the operation, you can write this outside of the log expression as a scalar multiplier.

Common log and natural log are usually the only logs that can be calculated directly on the calculator. The change of base formula can be used for calculating logs with other bases. It is also important to note other logarithmic properties. If the argument of the log and the base are the same, the expression is equal to 1. The log of 1, no matter the base, equals zero.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

Formulas to Know
Change of Base Property of Logs

log subscript b left parenthesis x right parenthesis equals fraction numerator log subscript a left parenthesis x right parenthesis over denominator log subscript a left parenthesis b right parenthesis end fraction

Other Properties of Logs

log subscript b left parenthesis b right parenthesis equals 1

log subscript b left parenthesis 1 right parenthesis equals 0

Power Property of Logs

log subscript b left parenthesis x to the power of n right parenthesis equals n times log subscript b left parenthesis x right parenthesis

Product Property of Logs

log subscript b left parenthesis x y right parenthesis equals log subscript b left parenthesis x right parenthesis plus log subscript b left parenthesis y right parenthesis

Quotient Property of Logs

log subscript b open parentheses x over y close parentheses equals log subscript b left parenthesis x right parenthesis minus log subscript b left parenthesis y right parenthesis

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