​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: 

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:

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

​The Difference Rule for Logarithms

For positive numbers x, y, and base b with b not equal to 1: 

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​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:

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

​The Power Rule for Logarithms

For positive numbers x, y, and base b with b not equal to 1: 

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

Expanding a Logarithm
To expand the log, bring the exponent out to the front.

​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.