This question describes exactly what I need to do!
I am also doing this using electronic circuitry. The problem is, that the answer (which has been marked as a 'solution') only works for those two specific numbers.
I simply want to concatenate the base 10 representation of two base 2 numbers together. For example: $(0001)_2+(0010)_2=(1100)_2$ if instead of the plus sign, there were a sign for the operation I want to perform, of course. I need some sort of formula that can do this for any 2 numbers.
Hopefully somebody can help me with this, since the internet has, yet again, let me down! Thank you in advance!!!
The short answer is the base $b$ concatenation of integers $x\geq 0$ and $y>0$ is given by $$x\cdot b^{\lfloor\log_by\rfloor +1} + y$$
From here, we can see it ultimately comes down to computing $\lfloor\log_by\rfloor$. I've never personally looked up how accurately we can compute said logarithms, so I would just preload integral powers of $b$ in whatever base $x,y$ appear in and then compare powers $b^i$ with $y$.
Find $b^n$ such that $b^n \leq y < b^{n+1}$. Then the concatenation is $$xb^{n+1}+y$$ and the calculation is done in whatever base is natural.