killing mushrooms

Hey all, I have several patches of mushrooms growing a round an older birch tree in my back yard. My mutt seems to try to eat them when he gets a chance. I have tried the dish soap/water solution, and now dish soap/baking soda/water solution. Weed killer spray does not seem to work. I the tree is staring to die I think...ergo all the mushrooms around it's base. I also dig them up as soon as I see them.

Any suggestions?