I missed a lesson on the product and quotient laws for logarithms and am having a little trouble figuring out the restrictions.

For example:

Log (x^2+7x+12) - Log (x^2-9)

= Log (x+3)(x+4) - Log (x-3)(x+3)

= Log [(x+3)(x+4)]/(x-3)(x+3)]

= Log (x+4)/(x-3)

How would I go about finding the restrictions on this logarithm?

I know in total, there are three possibilities: x>-3, x>-4, x>3

And in the back of the book, it shows that x>3 is the only restriction.

What I tried was to plug in each restriction into each separate log to see if it worked, but when I tried it that way I found that both x>3 and x>-4 worked, and that conflicts with the answer in the back of the book.

All help greatly appreciated!