I want to prove that the class of all Turing machines that use a logarithmic amount of space is closed under concatenation. The basic idea of my proof is this: given a word, to check if it's in $L_1L_2$, simply look at all possible $n+1$ ways to slice the word into 2 parts and check if the first part belongs to $L_1$ and if the second belongs to $L_2$.
Because both of the languages can be decided in logarithmic space, checking one after the other can also be done in logarithmic space.
Is my idea correct? Thanks.