I am not a mathematician.
I have some number $A_b$ which has a digit in a position $p$. That number can be written in another base as $V_d$ and it will have a range of digits $q_{start}$ – $q_{end}$ which hold (potentially added with other values) data from that digit.
I need to find the range $q_{start}$ – $q_{end}$.
And I can do a part of that for non-negative positions $p$ (to the left of the point) we know that the smallest non-zero digit in any base is $1$, thus $q_{start}$ is simply $\log_q\left(b^{p}\right)$ (round down), and $q_{end}$ is somewhat trivial once we bound it to $1$ in the next position, so it becomes $\log_q\left(b^{p+1}\right)$ (round up?).
However, I get completely lost once we approach digits to the right of the point. I know that in some situations $q_{end}$ will be infinite. I understand that it cannot happen when $d$ is a multiple of $b$. But somehow I am stuck.