To solve this logarithmic equation, we can use properties of logarithms to simplify the equation before isolating the variable x.

First, we can simplify the right side of the equation:
1/2 log6(x-1)^2 = log6((x-1)^2)^(1/2)
1/2 log6(x-1)^2 = log6|x-1|

Next, we can rewrite the left side of the equation using the properties of logarithms:
1+log6((x+3)/(x+7)) = log6(6) + log6((x+3)/(x+7))
1+log6((x+3)/(x+7)) = 1 + log6((x+3)/(x+7))

Now, we can rewrite the equation with the simplified expressions:
1 + log6((x+3)/(x+7)) = log6|x-1|

Since the bases are the same, we can drop the logarithm and solve for the expression inside the logarithm on each side of the equation:
(x+3)/(x+7) = |x-1|

Now, we can solve this equation for two cases:
Case 1: (x+3)/(x+7) = x-1
(x+3) = (x+7)(x-1)
x+3 = x^2 + 6x - 7
0 = x^2 + 5x - 10

Using the quadratic formula, we find that the solutions for x in this case are x = -2 and x = 5. However, we need to check these solutions in the original equation to see if they are valid.

Case 2: (x+3)/(x+7) = -(x-1)
(x+3) = -(x-1)(x+7)
x+3 = -x^2 - 6x + 7
x^2 + 5x + 4 = 0

Using the quadratic formula, we find that the solutions for x in this case are x = -4 and x = -1. As before, we need to check these solutions in the original equation to see if they are valid.

After checking all solutions in the original equation, we find that x = 5 is the only valid solution.