Show that k-clique lies in L
If you consider $k$ a constant, then k-clique can be solved be enumerating all $O(n^k)$ sets of vertices and checking that there are edges present in polynomial time (since the exponent is constant). 

Enumerating k-Tuples of numbers up to n can be done in $k\log n$ space, you only have to figure out how to check for the presence of an edge between two vertices without using much space. Depending on how much details you have to give this is either obvious or very tedious to write down.