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:
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
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