Proving language in Space Complexity
Here is a rough idea how to solve this. You compare $f(x)$ and $f(y)$ character by character. Your tape is devided into 4 pieces. The first one is for simulating  $f(x)$ the second one for $f(y)$, then you have two parts, one that counts which character of $x$ you are currently computig, and another $O(\log n)$ area that helps you to navigate. Now the programm looks like this


Determine $|x|$ and $|y|$ (store it in binary!)  and check if there are the same.
Compute the first character of $f(x)$ by simulating it on your machine. Store the output in a TM-state. Pause the Computation of $f(x)$.
Walk $|x|+1$ step to the right.
Simulate $f(y)$ until you get the first character, check if it coincides with the start of $f(x)$, if no reject, otherwise continue.
Walk $|x|$ steps left. 
Repeat the steps 2.-5. until you check $f(x)=f(y)$ completely.