The answer to the first question is
x 1 2 3 4 5 6 7 8 9 10
ilog(x) 0 1 4 2 6 5 9 3 8 7
The multiplication table is as follows:
1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 5 1 3 7 9 3 6 9 5 1 7 8 10 4 2 4 8 5 3 9 10 2 6 1 7 5 10 1 9 4 2 6 7 3 8 6 5 7 10 2 1 3 9 8 4 7 1 8 2 6 3 9 4 10 5 8 3 10 6 7 9 4 5 2 1 9 7 4 1 3 8 10 2 5 6 10 9 2 7 8 4 5 1 6 3I made a program to calculate ilog(x) and its multiplication table in any order N. The solutions (N=2...199) calculated by the program are given in the following files. To display the results of large N, I used a graphic expression in which each number is expressed by the gray scale colors (0 corresponds black, and N, white). The most intriguing thing is that there is no solution in N=195. I do not know the reason, but it is proven by my program which scans all of the cases.