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3 Tutorials that teach Properties of Logs
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Properties of Logs

Properties of Logs

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Properties of Logs

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Tutorial

  • Product Property
  • Quotient Property
  • Power Property
  • Change of Base
  • Other Properties

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.  In this lesson, we will be introduced to several logarithmic properties that arise when multiplication, division, and powers exist within logarithms. 

Product Property of Logs

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:


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, for example:


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:


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 connects individual logs with addition, and the quotient property connects individual logs with subtraction. 


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:

We can also apply this property in the other direction:


Power Property of Logs

The power property 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:

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

 As we see in the previous 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.

Change of Base

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:


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:


As we can see, log3(40) on its own is difficult to evaluate using a calculator.  However, thanks to the Change of Base formula, we can enter log(40) / log(3) into the calculator to evaluate the expression. 

Other Logarithmic Properties

Finally, let's review two other properties of logs that can help us simplify expressions.  The first states that if the argument of the log and the base are the same, the expression evaluates to 1. 

log subscript b left parenthesis b right parenthesis space equals space 1

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

log subscript b left parenthesis 1 right parenthesis space equals space 0