Practice for properties of logarithms

How You Can (And Should) Get More Practice!

Below is a few practice problems of various difficulty, but you will need considerably more practice than one each. For that reason you should definitely use the green “Try Another” button in the top right corner at least two or three times to complete additional versions of these questions for more practice. You should keep using that button until doing these problems feels straight forward and easy, and then come back after a week or so of doing other stuff and try again to make sure it is still just as easy for you.

Theoretically Easier Difficulty Problem

Use the properties that let you separate logarithms to split each factor into its own logarithm.
Use the property to pull down any powers of factors and make them into coefficients.
Expand the following logarithmic expression.

Use the property to transform any coefficients of logs into a power of the argument.
Use the property and/or to merge separate logarithms into one logarithm.

Condense the following logarithmic expression down to one log term.

Theoretically Medium Difficulty Problem

Use the properties that let you separate logarithms to split each factor into its own logarithm.
Use the property to pull down any powers of factors and make them into coefficients.

Expand the following logarithmic expression.

Use the property to transform any coefficients of logs into a power of the argument.
Use the property and/or to merge separate logarithms into one logarithm.

Condense the following logarithmic expression down to one log term.

Theoretically Harder Difficulty Problem

Use the properties that let you separate logarithms to split each factor into its own logarithm.
Use the property to pull down any powers of factors and make them into coefficients.

Expand the following logarithmic expression.

Use a change of base formula to convert all bases to the same value. Note that you will likely need to explicitly compute the log in the denominator and then rewrite it as a coefficient.
Use the property to transform any coefficients of logs into a power of the argument.
Use the property and/or to merge separate logarithms into one logarithm.

Condense the following logarithmic expression down to one log term.