Is there a law that you must have tailgate on a dump truck if you are hauling logs?

It's called a load securement law. If you're hauling loose logs in the back of a dump truck without a tailgate, it's an unsecured load, plain and simple. The state MIGHT allow the logs to be secured by straps, chains, or a combination of both, but if you're expecting to throw one strap and belly wrap the whole load with it, that won't suffice - if you read FMCSR 393.100 and up, you'll note that the DOT requires that load securements be rated for at least 50% of the weight of the load. A four inch strap going only over the top of the load has a Working Load Limit of 5400 lbs - when it's belly wrapped, that gets reduced by 50%. A two inch strap has a working load limit of 3300 lbs, and this is again halved when it's used as a belly wrap.