How to convert between bases partially?
Partial answer: 

It works for some pairs $b_1$ $b_2$, but may not work for every pair.

Partial solution:

for pairs $b_1$ $b_2$ where there is an $x, p_1, p_2 \in \mathbb{Z}$ such that 

$\begin{alignat*}{4}
x^{p_1} &amp; {}={} &amp; b_1\\
x^{p_2} &amp; {}={} &amp; b_2
\end{alignat*}$

you can divide $n$ of $b_1$ into sections of $p_2 $ digit sections, which you can then convert into $b_2$.

If you want to query a slice in the result $b_2$, you need to calculate the section number by dividing by $p_1 $

3 -> 10 doesn't work with this, but I will give you an example 

8 -> 16:

$x = 2, p_1 = 3, p_2 = 4$

8  [....][1][2][3][4][..p2 * x blocks..]
2        001010011100
16 [....][ 2][ 9][ C][..p1 * x blocks..]


3 -> 9:

$x = 3, p_1 = 1, p_2 = 2$

3  [....][1][0][2][1][..p2 * x blocks..]
3          1  0  2  1
9  [....][   3][   7][..p1 * x blocks..]