Hi there, can you post pics of him + of his brown vent ? That wouldn't be caused by slipping off a log, but if he hurt his spine it could affect his ability to poo if he has a spine injury, but that would be evident. What did the vet tell you to do or give you for him ? Some natural laxatives that usually work are any of these or a combo....baby food squash or prunes, applesauce, mixed with about 1/4 tsp. of veg oil [ or just a few drops for a juvenile beardie. ] You can also try this right away, an extra warm [ more than usual ] shoulder height
bath. Stay with him of course, but try to get him to swim a bit in the warm water.
What exact type of lights do you use and what are the temps. in the tank ? Post a few pics of his set up if you can, but esp. of his vent. Here's how:
https://www.beardeddragon.org/useruploads/ Then hit XIMG