It figures, 5 minutes after I post I get it.
The problem is that I was ignoring another property of logarithms. log_6(x+3) + log_6(x+4) = 1 is the same as log_6(x+3)(x+4) = 1. To find X, I set (x+3)(x+4) = 6, and solve. This factors down to (x+1)(x+6), which is x=-1, x=-6. Because you cannot take a negative logarithm, x=-6 is thrown out as it breaks BOTH logarithms, so the answer is -1.