IXL - Properties of logarithms: mixed review (Precalculus practice)

Expand the logarithm. Assume all expressions exist and are well-defined.

Write your answer as a sum or difference of base-2 logarithms or multiples of base-2 logarithms. The inside of each logarithm must be a distinct constant or variable.

log

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question

Expand the logarithm. Assume all expressions exist and are well-defined.

Write your answer as a sum or difference of common logarithms or multiples of common logarithms. The inside of each logarithm must be a distinct constant or variable.

log

key idea

Product Property:

log

where

and

are positive, and

≠

Quotient Property:

log

–

log

where

and

are positive, and

≠

Power Property:

log

where

and

are positive, and

≠

These properties follow an order of operations:

When
expanding,
use the Quotient and Product Properties first, then the Power Property last.
When rewriting as a single
logarithm,
use the Power Property
first,
then the Product and Quotient Properties last.

A common logarithm uses base 10. The common logarithm

log

is usually written as

log

without the base.

solution

Expand the logarithm.

log

Power Property

log

Power Property

You could also use the Product Property.

log

Definition of exponent

log

(

)

Definition of exponent

Product Property

log

Product Property

Combine like terms

log

Combine like terms

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Not feeling ready yet? These can help:

I.4Product property of logarithms

I.5Quotient property of logarithms

I.6Power property of logarithms

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