Without seeing the actual design it will be hard to tell, but one thing is for sure, it should be watertight without having to use a paint product over the wood.
The one I built was probably a bit bigger than yours ( 180 M^2) but the principle are the same. Did you use waterproof gaskets between the logs ? How hard did you bray the logs together and is there threaded rods to tighten to close up any gaps ?
I dont like to be negative about this kind of stuff as there is always a way to fix this sort of thing but it may mean taking the roof off and re doing the leaking logs. another alternative would be to use a membrane on the inside ( Tyvek or similar) and then re clad the inside, doing this you would need to prevent as much water getting the join as possible, but the logs would be able to control the moisture fairly well and you would have a dry space inside.