SOLUTION: Solve for x: log(x-3)= -log (x) + log (2) Answer Choices: A) 5/2 B) (3+sqrt(17))/2 C) sqrt(6) D) {1,2}

Algebra ->  Trigonometry-basics -> SOLUTION: Solve for x: log(x-3)= -log (x) + log (2) Answer Choices: A) 5/2 B) (3+sqrt(17))/2 C) sqrt(6) D) {1,2}      Log On


   

Question 253703: Solve for x: log(x-3)= -log (x) + log (2)
Answer Choices:
A) 5/2 B) (3+sqrt(17))/2 C) sqrt(6) D) {1,2}
Answer by edjones(8007) About Me  (Show Source):
You can put this solution on YOUR website!
log(x-3)= -log (x) + log (2)
10^(log(x-3))=10^(-log(x))+10^(log(2)) Raising logs base 10 to the 10th power cancels them out.
x-3=1/x * 2 Sign changes from a + to a * because adding logs is multiplying.
x-3=2/x
x^2-3x=2 multiply each side by x.
x^2-3x-2=0
x=(3+sqrt(17))/2 see below [x=(3-sqrt(17))/2 is not an answer because logs of negative numbers are not allowed.]
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This was a tough one; don't give up; you will get it!
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Ed
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Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-3x%2B-2+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-3%29%5E2-4%2A1%2A-2=17.

Discriminant d=17 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--3%2B-sqrt%28+17+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-3%29%2Bsqrt%28+17+%29%29%2F2%5C1+=+3.56155281280883
x%5B2%5D+=+%28-%28-3%29-sqrt%28+17+%29%29%2F2%5C1+=+-0.56155281280883

Quadratic expression 1x%5E2%2B-3x%2B-2 can be factored:
1x%5E2%2B-3x%2B-2+=+1%28x-3.56155281280883%29%2A%28x--0.56155281280883%29
Again, the answer is: 3.56155281280883, -0.56155281280883. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-3%2Ax%2B-2+%29