Does checking the middle bit of an input require a log(n) space pointer?
You can do this with constant amount of memory (according to your definition of space complexity). Assume $n \ge 2$ (the complementary case is trivial). 
Mark the work tape on the starting position of the input, then move right until you reach the end of the input and mark the last position of the input on the work tape.

Repeat the following:


Go back to towards the beginning of the input until you find a marked position on the work tape. 
If the position immediately to the right is already marked, report the current position as the middle of the input.
Otherwise, remove the mark from the current position (i.e., write the empty symbol), move right, and mark the current position.
Advance towards the end of the input until you find a marked position on the work tape.
If the position immediately to the left is already marked, report that position as the middle of the input.
Otherwise, remove the mark from the current position, move left, and mark the current position.