SOLUTION: Alright... I've completely forgotten how to do questions like this. The expression 4logx−5log(x^2+1)+4log(x−1) can be written as a single logarithm log(A). A= ???

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Alright... I've completely forgotten how to do questions like this. The expression 4logx−5log(x^2+1)+4log(x−1) can be written as a single logarithm log(A). A= ???      Log On


   

Question 220838: Alright... I've completely forgotten how to do questions like this.
The expression 4logx−5log(x^2+1)+4log(x−1) can be written as a single logarithm log(A).
A= ????
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
You'll need three of the basic properties of logarithms:
  • log%28a%2C+%28b%5Ec%29%29+=+c%2Alog%28a%2C+%28b%29%29 This can be use to move a coefficient in front of a log and "move" it into the argument as an exponent and vice versa.
  • log%28a%2C+%28b%29%29+%2B+log%28a%2C+%28c%29%29+=+log%28a%2C+%28b%2Ac%29%29 This allows us to "add" two logarithms resulting in a log of the product of the two arguements. It also works backwards. You can "split" the log of a product into the sum of the logs of the factors.
  • log%28a%2C+%28b%29%29+-+log%28a%2C+%28c%29%29+=+log%28a%2C+%28b%2Fc%29%29 This allows us "subtract" logs or to "split" the log of a quotient into the difference of the logs of the numerator and denominator.

Note that the last two properties do not have coefficients in front of the logs. So we need to use the first property to eliminate any coefficients before we try to use the other two properties.

So with 4%2Alog%28%28x%29%29-5%2Alog%28%28x%5E2%2B1%29%29%2B4%2Alog%28%28x-1%29%29 we need to start by "moving" the coefficients into the arguments as exponents using the first property above:

Now we can use the third property to "subtract" the first two logs:

And finally we can use the second property to "add" the two logs: