Showing that Independent set of size $k$ can be decided using logarithmic space
You can use the following algorithm.

I assume that the vertices are labeled $1,\ldots,m$. You enumerate all $k$-tuples $\{1,\ldots , m\}^k$. This can be done by reserving $k$ counters on the TM tape. Since $k$ is not part of your input this needs only $O(\log m )\subseteq O(\log n)$ space. However for every $k$-tuple you can test if it is an independent set. Simpy test all pair of selected vertices if there are not connected by an edge. Again, since $k$ is fixed the edges you have to test (depending on the current tuple) are known. Thus this test can be hard-wired into the TM program, and therefore needs no extra-space.