Sigma wrote:the professional framers locally use 3/4" masking tape
Any professional framer worth his/her salt would not use masking tape but I know some do and it is really bad practice. As mentioned, gummed tape is probably the best solution although it can be harder to work with than a self adhesive tape - until you get the knack. I would recommend Eco 25+ tape from Lion and have been using this for 10 years.
The Logan (not sure which version you have) should be capable of good cuts but a common mistake is to have the blade depth set too deep. It should only slightly protrude into the scrap mat and is worth experimenting to get this right. Lower end cutters do not have the blade supported as well as more expensive models. One solution is to not cut through the board completely when starting the cut, instead just do say 1mm for 20mm - return to the start of the cut and cut deeper - do this again for the final cut and then the full length in one go. This is easier if the Logun clamps the cutting bar to the mount.
Also change the blade very frequently. I use a Keencut Ultimat Gold and change the blade after every 4 cuts, my blades are reversible so that keeps the cost down but blades are cheaper than mountboard.