SOLUTION: Write the expression in terms of simpler logarithmic forms. ln x^4y^2/z^1/2

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Write the expression in terms of simpler logarithmic forms. ln x^4y^2/z^1/2      Log On


   

Question 368653: Write the expression in terms of simpler logarithmic forms.
ln x^4y^2/z^1/2
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
ln%28%28x%5E4y%5E2%29%2Fz%5E%281%2F2%29%29
There are three properties of logarithms which can be used to manipulate them:
  • log%28a%2C+%28p%2Aq%29%29+=+log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29
  • log%28a%2C+%28p%2Fq%29%29+=+log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29
  • log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29

Since the argument of your logarithm is a division/fraction, we will use the second property to split it into two logarithms:
ln%28x%5E4y%5E2%29+-+ln%28z%5E%281%2F2%29%29
The argument of the first logarithm is a product so we will use the first property to split it into two logarithms:
ln%28x%5E4%29+%2B+ln%28y%5E2%29+-+ln%28z%5E%281%2F2%29%29
The arguments all three logarithms have an exponent. We can use the third property on them to move the argument out in front:
4%2Aln%28x%29+%2B+2%2Aln%28y%29+-+%281%2F2%29%2Aln%28z%29
By using the properties as we have, the expression as a whole is more complex but the arguments of the logarithms are now just a simple variable.