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Logarithm Rules


​Rules for Logarithms With the Same Base

​Just as there are rules for working with exponents, there are also rules you can use when working with logarithms.  These rules can be used to simplify or expand logarithms.  In this lesson, we'll look at 3 rules: the Sum Rule, the Difference Rule, and the Power Rule.  Keep in mind that these rules only apply for logarithms with the same base.

The Sum Rule for Logarithms

For positive numbers x, y, and base b with b not equal to 1: 
Sum of Two Logs Rule
Make sure to notice that the base stays the same.  This rule only applies to logarithms with the same base.

Simplifying a Logarithm
This rule allows you to simplify two logs that are being added and rewrite it as a single logarithm:
How to simplify the sum of two logarithms.

Expanding a Logarithm
This rule also allows you to expand (split up) a logarithm into two separate logs.
How to use the sum rule to expand a logarithm


​The Difference Rule for Logarithms

For positive numbers x, y, and base b with b not equal to 1: 
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Difference of Two Logarithms Rule

​Make sure to notice that the base stays the same.  This rule only applies to logarithms with the same base.
​
Simplifying a Logarithm
This rule allows you to simplify two logs that are being subtracted and rewrite it as a single logarithm:
How to simplify a logarithm with the difference rule.

Expanding a Logarithm
This rule also allows you to expand (split up) a logarithm into two separate logs.
How to expand a log using the difference rule.


​The Power Rule for Logarithms

For positive numbers x, y, and base b with b not equal to 1: 
Power Rule for Logarithms

​Make sure to notice that the base stays the same.  This rule only applies to logarithms with the same base.
​
Simplifying a Logarithm
How to simplify a logarithm using the power rule.

Expanding a Logarithm

To expand the log, bring the exponent out to the front.
How to expand a logarithm using the power rule.


​Logarithm Rules Video

Want to see a few more examples? Check out the short video below.


Practice

Ready to try some problems on your own? Click the START button to try a practice quiz.

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Want to learn how to solve an equation with logarithms?  Check out one of the solving logarithmic equations lessons.
Want to try one of my digital math activities for free?  Click the link to grab the teacher version or student version.

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