importantly this shows thatbecause the cancels out the number of operations at each level is independent of the level this gives us an upper bound to the number of operations carried out on each levelin this example it should be pointed out that this includes the number of operations performed by each recursive call on that levelnot the recursive calls made on subsequent levels this shows that the work doneas the number of recursive calls doubles with each levelis exactly counter balanced by the fact that the input size for each sub problem is halved to find the total number of operations for complete merge sort we simply multiply the number of operations on each level by the number of levels this gives us the followingwhen we expand this outwe get the followingthe key point to take from this is that there is logarithmic component to the relationship between the size of the input and the total running time if you remember from school mathematicsthe distinguishing characteristic of the logarithm function is that it flattens off very quickly as an input variablexincreases in sizethe output variabley increases by smaller and smaller amounts for examplecompare the log function to linear function