Complexity of $n \times \log n$ tiling problem
Hint: Use dynamic programming, going row by row. Use the fact that there are only $\log n$ many columns, and so only $|A|^{\log n} = n^{\log |A|}$ many different configurations of each column.

(The resulting algorithm will be polynomial in $n$ but not in the input size, which is only $O(\log n)$.)