Question 31386
To solve this, you would use the "quotient" rule for logarithms rather than the "product" rule for logarithms.
The quotient rules states:
{{{log(M) - log(N) = log((M/N))}}} Applying this rule to your problem, we have:
{{{log((3+x)) - log((x-3)) = log(((3+x)/(x-3)))}}} and this = {{{log((3))}}}, so:
{{{log(((3+x)/(x-3))) = log((3))}}} Therefore:
{{{((3+x))/((x-3)) = 3}}} Now you can solve for x.  Multiply both sides by (x-3)
{{{3+x = 3(x-3)}}} Simplify.
{{{3+x = 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((3+6))-log((6-3)) = log((9))-log((3))}}} Applying the "quotient" rule, we get:
{{{log((9))-log((3)) = log((9/3))}}} = {{{log((3))}}}