Why is the unary representation of a number exponentially larger than a base k representation of it?
The base $k$ representation of the number $n$ takes $\log_{k}n$ bits, whereas the unary representation takes $n$, and of course $n = k^{\log_{k}n}$

The gap is more apparent with larger numbers, if we take our normal base 10 system, then writing 1000 takes 4 decimal digits, but writing it in unary takes 1000 ones.