Installing plumbing in a log home is pretty much the same as a conventionally built home. If you have interior framed walls your plumber will run all required plumbing there.
If you have all log home interior walls the contractor will have to build a chase-way in one of the rooms on your first level. This chase-way is a boxed out area to run plumbing and HVAC through to the second level of your log home.
If you have chosen to use full round log floor joists on the second level of your log home you will have to build the floor up to allow for plumbing and electrical. This is done by taking 2”x6” boards and standing them on end to create a void between the tongue and grove most people install on top of their floor joists. Then the log home contractor will install ¾” OSB or plywood for the floor.