A.
$n^{\lg c} = n^k $ has polynomial growth since $ k = \lg c $ is a constant.
$c^{\lg n}$ is exponential in $\lg n$ and hence $O(n).$
So, $$n^{\lg c} = \Theta\left(c^{\lg n}\right).$$
B.
$\lg (n^n) = n \lg n$
$\lg (n!) = \Theta ( n \lg n)$ (Stirling's approximation)
So, $$\lg (n!) = \Theta(\lg n^n).$$