Here is an infinite set of numbers
Suppose a logarithm function which satisfy
on the modulo N and the function is one-to-one, thus
The range of the function 
must be in the set
of numbers
We call the set of the above conditions "cyclic logarithm condition".
For example in the case of N=4 (thus 
),
assuming the following multiplication table
,
the function
satisfies the cyclic logarithm condition.
(E.g. according to the multiplication table,
and their logarithms have the following relation
