SOLUTION: write log(4(x^1/4))/log(y^3) as a sum, difference, and/or multiple of log 4, log x, and log y

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Question 815650: write log(4(x^1/4))/log(y^3) as a sum, difference, and/or multiple of log 4, log x, and log y
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
To do problems like this, we will use some or all of the following properties:
  1. log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29
  2. log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29
  3. log%28a%2C+%28p%5En%29%29+=+n%2Alog%28a%2C+%28p%2Aq%29%29
Note: For this type of problem we will be using these properties from right to left.

log%28%284%28x%5E%281%2F4%29%29%29%29%2Flog%28%28y%5E3%29%29
The argument of the log in the numerator is a product. For this we will use the first property to split it into two logs:
%28log%28%284%29%29%2Blog%28%28x%5E%281%2F4%29%29%29%29%2Flog%28%28y%5E3%29%29
The second log in the numerator and the log in the denominator have exponents on their arguments. We will use the third property to move the exponents out in front:
%28log%28%284%29%29%2B%281%2F4%29log%28%28x%29%29%29%2F3log%28%28y%29%29
This may be an acceptable answer. All the logs are logs of 4, x or y. But it does involve a quotient which your instructions did not include. There's no way to eliminate the quotient so I hope you just forgot to include it.

The expression does have a fraction, 1/4, within a larger fraction. This is not always an acceptable form for an answer. We could get a better form for the answer by multiplying the numerator and denominator by 4:
%28%28log%28%284%29%29%2B%281%2F4%29log%28%28x%29%29%29%2F3log%28%28y%29%29%29%284%2F4%29
Multiplying (using the Distributive Property on top) we get:
%284log%28%284%29%29%2Blog%28%28x%29%29%29%2F12log%28%28y%29%29