SOLUTION: Write the expression as a single logarithm. Express powers as factors. (Simplify your answer) ln(x/(x-2))+ln((x+2)/x)-ln((x^2)-4)

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Write the expression as a single logarithm. Express powers as factors. (Simplify your answer) ln(x/(x-2))+ln((x+2)/x)-ln((x^2)-4)      Log On


   

Question 365204: Write the expression as a single logarithm. Express powers as factors. (Simplify your answer)
ln(x/(x-2))+ln((x+2)/x)-ln((x^2)-4)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
I assume the expression is:
ln%28x%2F%28x-2%29%29%2Bln%28%28x%2B2%29%2Fx%29-ln%28x%5E2-4%29
These are not like terms because the arguments are different. But there are properties of logarithms which allow us to combine these logarithms into one:
  • log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29
  • log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29

With these properties all you need is the same base and coefficients of 1 and your logarithms meet both requirements. Using the first property on the first two logarithms (because of the "+" between them) we get:
ln%28x%2F%28x-2%29%29%2A%28%28x%2B2%29%2Fx%29-ln%28x%5E2-4%29
The x's cancel:
ln%28cross%28x%29%2F%28x-2%29%29%2A%28%28x%2B2%29%2Fcross%28x%29%29-ln%28x%5E2-4%29
leaving:
ln%28%28x%2B2%29%2F%28x-2%29%29-ln%28x%5E2-4%29
Now we can use the second property (because of the "-" between them):
ln%28%28%28x%2B2%29%2F%28x-2%29%29%2F%28x%5E2-4%29%29
Now we try to simplify the compllex fraction. We start by factoring the denominator:
ln%28%28%28x%2B2%29%2F%28x-2%29%29%2F%28%28x%2B2%29%28x-2%29%29%29
Now we will multiply the numerator and denominator by (x-2):

The (x-2)'s in the numerator cancel:

leaving:
ln%28%28%28%28x%2B2%29%2F1%29%2F%28%28x%2B2%29%28x-2%29%29%29%281%2F%28x-2%29%29%29
which simplifies to:
ln%28%28x%2B2%29%2F%28%28x%2B2%29%28x-2%29%28x-2%29%29%29
Now the (x+2)'s cancel:
ln%28cross%28%28x%2B2%29%29%2F%28cross%28%28x%2B2%29%29%28x-2%29%28x-2%29%29%29
leaving:
ln%281%2F%28%28x-2%29%28x-2%29%29%29
Since it says not to use powers, we will not change the denominator to %28x-2%29%5E2. This may be an acceptable answer. However we can simplify a little further. We can rewrite the argument as follows:
ln%28%28%28x-2%29%28x-2%29%29%5E%28-1%29%29
and then use the thrid property of logarithms,
or
-ln%28%28x-2%29%28x-2%29%29%29