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

Question 31386: Solve the equation:
log(3+x)-log(x-3)=log 3
Found 2 solutions by mukhopadhyay, Earlsdon:
Answer by mukhopadhyay(490) (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) (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:
Applying this rule to your problem, we have:
and this = , so:
Therefore:
Now you can solve for x. Multiply both sides by (x-3)
Simplify.
Subtract x from both sides.
Add 9 to both sides.
Finally, divide both sides by 2.

Solution is:
x = 6
Check:
Applying the "quotient" rule, we get:
=
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