Reductions between languages of different densities?
Let $A$ be any language not in $L$, such that $A$ has density $2^{o(n)}$, and define 
$$B = \{s \circ 1 | s \in \{0,1\}^*\} \cup \{s \circ 0 | s \in A\}.$$
Here $\circ$ is concatenation. The language $B$ has density $\Omega(2^n)$, which is superpolynomial in $2^{o(n)}$. On the other hand, $A$ and $B$ log-space reduce to each other ($A$ to $B$ by concatenating $0$, and $B$ to $A$ by reducing all strings ending in $1$ to the smallest yes instance of A, and removing the last bit from all strings ending in $0$). Hence $B \notin L$ as well.