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.
Sum of Two 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.
Difference of Two 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.
Logarithm of a Power
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.
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Want to learn how to solve an equation with logarithms? Check out one of the solving logarithmic equations lessons.