SOLUTION: Use logarithmic properties to expand the expression,ln(e^3y/x^2),as much as possible. I would appriciate some help.

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Question 201782: Use logarithmic properties to expand the expression,ln(e^3y/x^2),as much as possible. I would appriciate some help.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
There are four properties which allow us to manipulate logarithmic expressions. The first one allow you to change bases. (Since the software on Algebra.com doesn't support writing logarithms legibly, I'm going to use l%5Ba%5D to represent "log base a".) The other three require that the bases of the logarithms being manipulated be all the same. (Since this problem involves ln I will express these last three properties using ln.) The properties are:
  1. l%5Ba%5D+%28x%29+=+%281%2F%28l%5Bb%5D+%28a%29%29%29%2Al%5Bb%5D+%28x%29 allows you to change the base from a to b. (Ignore the multiplication (*) between the l%5Ba%5D and the %28x%29 and between the l%5Bb%5D and the %28x%29. Another limitation of Algebra.com)
  2. ln%28x%5Ea%29+=+a%2Aln%28x%29 This allows you to take an exponent from the argument to ln and move it in front as a coefficient of ln. You can also use it the other way to move a coefficient from in front of ln to the exponent of ln.
  3. ln%28x%2Ay%29+=+ln%28x%29+%2B+ln%28y%29 Used in one direction we can separate the factors of an argument to ln. In the other direction it allows you to "add" ln's with different arguments.
  4. ln%28x%2Fy%29+=+ln%28x%29+-+ln%28y%29 In one direction we can separate the numerator and denominator of a fraction in the argument of ln. In the other direction we can "subtract" ln's with different arguments.


Since your problem is all ln's we will not need property #1 to change bases. To simplfy
ln%28%28e%5E%283y%29%29%2F%28x%5E2%29%29
We'll start by using property #4 to separate the numerator and the denominator:
ln%28e%5E%283y%29%29+-+ln%28x%5E2%29
Next we will use property #2 to move the exponents in the arguments to the front:
3y+%2A+ln%28e%29+-+2%2Aln%28x%29
Since ln(e) = 1 we get:
3y+-+2%2Aln%28x%29
Since there's nothing elase we can do to make the expression simpler, we are done.