The machine includes a feeding path for the logs to be cut and a cutting head arranged along the feeding path. The cutting head includes a coupling for a disc-shaped cutting blade and is designed to impart the disc-shaped cutting blade a rotary motion around the axis thereof and a cyclic movement to cut the logs into single rolls, and to allow the logs to move forward along the feeding path. The machine also includes at least a grinding unit, including at least one grinding wheel mounted onto a rotation shaft and co-acting with a side of the disc-shaped cutting blade to grind the cutting edge of the same disc-shaped cutting blade. A controlled approach system is also provided to move the grinding wheel towards the disc-shaped cutting blade.