SOLUTION: Question1: log(x-2)+log(x+1)=2 This is what I have so far.. However, I'm not confident it's correct. Please help. Thank you!! log[(x-2)(x+1)]=2 b=1 p=2 x=(x-2)(x+1)

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Question1: log(x-2)+log(x+1)=2 This is what I have so far.. However, I'm not confident it's correct. Please help. Thank you!! log[(x-2)(x+1)]=2 b=1 p=2 x=(x-2)(x+1)       Log On


   

Question 378300: Question1: log(x-2)+log(x+1)=2
This is what I have so far.. However, I'm not confident it's correct. Please help. Thank you!!
log[(x-2)(x+1)]=2
b=1 p=2 x=(x-2)(x+1)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log(x-2) + log(x+1) = 2
You are correct with
log((x-2)(x+1)) = 2
But I am confused with the "b=1 p=2 and x=(x-2)(x+1)"

The next step is to rewrite the equation in exponential form. In general log%28a%2C+%28p%29%29+=+q is equivalent to p+=+a%5Eq. Using this on our equation we get:
%28x-2%29%28x%2B1%29+=+10%5E2 (Since the base of log is 10.)
Simplifying each side we get:
x%5E2+-x+-2+=+100
This is a quadratic equation so we want one side to be zero. Subtracting 100 from each side we get:
x%5E2+-x+-+102+=+0
This will not factor but we can use the Quadratic Formula:
x+=+%28-%28-1%29+%2B-+sqrt%28%28-1%29%5E2+-+4%281%29%28-102%29%29%29%2F2%281%29
which simplifies as follows:
x+=+%28-%28-1%29+%2B-+sqrt%281+-+4%281%29%28-102%29%29%29%2F2%281%29
x+=+%28-%28-1%29+%2B-+sqrt%281+%2B+408%29%29%2F2%281%29
x+=+%28-%28-1%29+%2B-+sqrt%28409%29%29%2F2%281%29
x+=+%281+%2B-+sqrt%28409%29%29%2F2
In long form this is:
x+=+%281+%2B+sqrt%28409%29%29%2F2 or x+=+%281+-+sqrt%28409%29%29%2F2

When solving equations where the variable is in the argument (or base) of a logarithm, you must check you answers. You must make sure that all arguments (and bases) of logarithms are positive. If a "solution" makes an argument (or base) negative or zero then that "solution" must be rejected. This can happen even if no mistakes were made while finding the solution! This is why we must check out answers in problems like this.

When checking answers use the original equation:
log(x-2) + log(x+1) = 2
Checking x+=+%281+%2B+sqrt%28409%29%29%2F2:

Since the sqrt%28409%29+%3E+20 we should be able to see that both arguments will be positive. So there is no reason to reject this solution. The rest of the check is optional. It will determine if we made a mistake. I'll leave that up to you.

Checking x+=+%281+-+sqrt%28409%29%29%2F2:

Since the sqrt%28409%29+%3E+20 we should be able to see that both arguments will be negative! So we must reject this "solution".

So the only solution to your equation is:
x+=+%281+%2B+sqrt%28409%29%29%2F2