SOLUTION: Express the following as an equivalent expression using the individual logarithms of w, x, y, and z. Loga [ (x3 y ) / ( w2 z2 ) ]

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Question 292415: Express the following as an equivalent expression using the individual logarithms of w, x, y, and z.
Loga [ (x3 y ) / ( w2 z2 ) ]
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The following properties of logarithms are keys to problems like this:
  1. log%28a%2C+%28p%2Aq%29%29+=+log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29
  2. log%28a%2C+%28p%2Fq%29%29+=+log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29
  3. log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29

Let's see how these work on your expression:
log%28a%2C+%28%28x%5E3%2Ay%29%2F%28w%5E2z%5E2%29%29%29
Start with property #2 (because the argument is a big fraction/quotient):
log%28a%2C+%28x%5E3%2Ay%29%29+-+log%28a%2C+%28w%5E2z%5E2%29%29
Next, since each argument is a product, we'll use property #1. (Note the use of parentheses. Whenever you replace one term with multiple terms, it is important to sue parentheses.)

The first, third and fourth logarithms have exponents on the arguments. So we'll use property #3 on them:

And finally we simplify. (Note how the parentheses help us know that the subtraction in the middle applies to both terms that follow it.)