Does solving subproblems in logspace in different models implies solving original problem in logspace? Let's consider the following situation:

We have a problem $\mathsf{L\text{}}$ . We would like to solve it in $LOGSPACE$. Let assume that our problem $\mathsf{L\text{}}$  can be divided to two parts: $A$ and $B$.

We can solve $A$ using computatation model $M_1$ in $LOGSPACE$.

We can solve $B$ using computation model $M_2$ in $LOGSPACE$, $M_1 \neq M_2$

Does it mean that we  can solve $P$ in $LOGSPACE$ ? Why?
The composition of two logspace functions is again in logspace. This is a classical result that you can try to prove on your own.