SOLUTION: Solve the equation: log(3+x)-log(x-3)=log 3

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Question 31386: Solve the equation:
log(3+x)-log(x-3)=log 3
Found 2 solutions by mukhopadhyay, Earlsdon:
Answer by mukhopadhyay(490) About Me  (Show Source):
You can put this solution on YOUR website!
log(3+x)-log(x-3)=log 3
=>log[(x+3)(x-3)]=log 3
=>log (x^2-9)=log 3
=>x^2-9 = 3
=>x^2 = 12
=>x = sqrt(12) or x = -sqrt(12)
x cannot be anyone of them because log(x-3) and log(x+3) are valid as long as (x-3) is not negative and (x+3) is not negative.
(x-3) is negative if x = sqrt(12) or x = -sqrt(12)
So, the answer is x has a null set

Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
To solve this, you would use the "quotient" rule for logarithms rather than the "product" rule for logarithms.
The quotient rules states:
log%28M%29+-+log%28N%29+=+log%28%28M%2FN%29%29 Applying this rule to your problem, we have:
log%28%283%2Bx%29%29+-+log%28%28x-3%29%29+=+log%28%28%283%2Bx%29%2F%28x-3%29%29%29 and this = log%28%283%29%29, so:
log%28%28%283%2Bx%29%2F%28x-3%29%29%29+=+log%28%283%29%29 Therefore:
%28%283%2Bx%29%29%2F%28%28x-3%29%29+=+3 Now you can solve for x. Multiply both sides by (x-3)
3%2Bx+=+3%28x-3%29 Simplify.
3%2Bx+=+3x-9 Subtract x from both sides.
3+=+2x-9 Add 9 to both sides.
12+=+2x Finally, divide both sides by 2.
6+=+x
Solution is:
x = 6
Check:
log%28%283%2B6%29%29-log%28%286-3%29%29+=+log%28%289%29%29-log%28%283%29%29 Applying the "quotient" rule, we get:
log%28%289%29%29-log%28%283%29%29+=+log%28%289%2F3%29%29 = log%28%283%29%29