LOGSPACE in different Turing Machine variants
Let T(n) be the time complexity of problem on two tapes Turing machine then time complexity on single tape Turing machine is going to $O(T(n)^2)$ (more precisely $10 \times T(n)^2$). You have mentioned that your problem is in $LOGSPACE$, So $T(n)$ will be going to be some polynomial in $n$. To simulate two tapes Turing machine by single tape encode locations 1,3,5,7... to encode the first tape and locations 2,4,6... to encode second tape and also use symbol for the encoding of head positions. Now to simulate the one movement of two tape Turing machine you need to first find the head position (swap left to right) so you need $O(T(n))$ steps. Now to simulate the $O(T(n)$ many steps you need $O(T(n)^2)$ time.