SOLUTION: Compress the expression {{{2(ln(x-3)+ln(x))-ln(x^(2)-9)}}} into one simplified logarithm.

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Compress the expression {{{2(ln(x-3)+ln(x))-ln(x^(2)-9)}}} into one simplified logarithm.      Log On


   

Question 258258: Compress the expression 2%28ln%28x-3%29%2Bln%28x%29%29-ln%28x%5E%282%29-9%29 into one simplified logarithm.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
2%28ln%28x-3%29%2Bln%28x%29%29-ln%28x%5E%282%29-9%29
To combine logarithms we use two of the properties of logarithms:
  • 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

These properties require that the coefficients of the logarithms are 1's. For logarithms which have other coefficients, we have a third property which allows us to move a coefficient of a logarithm into the argument as an exponent: q%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5Eq%29%29
Inside the parentheses we find ln%28x-3%29+%2B+ln%28x%29. Since this is an addition, we can use the first property to combine these:
2%28ln%28%28x-3%29%2A%28x%29%29%29-ln%28x%5E%282%29-9%29
Next, we can use the third property to move the 2 from in front into the argument of the logarithm:
ln%28%28%28x-3%29%2A%28x%29%29%5E2%29-ln%28x%5E%282%29-9%29
And finally, since this is a subtraction, we can use the second property to combine the remaining logarithms:
ln%28%28%28%28x-3%29%2A%28x%29%29%5E2%29%2F%28x%5E%282%29-9%29%29
We have now condensed the expression into a single logarithm. This may be an acceptable answer. But we can simplify the argument of the logarithm. The fraction will reduce. After we factor the denominator we get:
ln%28%28%28x-3%29%2A%28x%29%29%5E2%29%2F%28%28x%2B3%29%28x-3%29%29
We can see that the x-3 in the denominator will cancel with one of the two (x-3)'s (it is squared, after all) leaving:
ln%28%28%28x-3%29%2Ax%5E2%29%2F%28x%2B3%29%29
Last of all we can multiply out the numerator:
ln%28%28x%5E3-3x%5E2%29%2F%28x%2B3%29%29