The scope of this paper is twofold. First, to emphasize the use of the mod 1 map in exploring the digit distribution of random variables. We show that the well-known base- and scale-invariance of Benford variables are consequences of their associated mod 1 density functions being uniformly distributed. Second, to introduce a new concept of the $n$-digit Benford variable. Such a variable is Benford in the first $n$ digits, but it is not guaranteed to have a logarithmic distribution beyond the $n$-th digit. We conclude the paper by giving a general construction method for $n$-digit Benford variables, and provide a concrete example.