Can CNF with an input string be evaluated in logarithmic space?
Yes, you can decide this language in logarithmic space, using the following algorithm:


for each clause in the CNF:


do a linear scan of $w$ to see if one of the literals in the clause is satisfied; if not, reject

if you got through all the clauses without rejecting, accept


What's the space complexity?  Well, there are at most $n$ clauses, so we need at most $\lg n$ bits to store the loop counter to keep track of which clause we're currently on.  We also need to store the current clause; there are at most $n$ variables in the formula, so we need at most $3 (1+\lg n)$ bits to store the clause.  Therefore, we need at most $O(\lg n)$ bits in total of space.

Of course, the running time is not optimal (it is $O(n^2)$ or so), but that doesn't matter, as we're only counting space complexity.