Asymptotic relation between $n^{lg (c)}$ and $c^{lg (n)}$?

Assume that $$c > 0$$ is a constant and that $$n$$ is a positive variable. Then why is $$n^{lg (c)} = O(c^{lg (n)})$$? Furthermore, can you explain why $$n^{lg (c)}$$ is not $$o(c^{lg (n)})$$, that is $$n^{lg (c)} \neq o(c^{lg (n)})$$?

Taking the lg of both gives $$\operatorname{lg}(c)\operatorname{lg}(n)$$ so they are actually equal. Hence the provided asymptotics follow from $$f\in\mathcal O(f)$$ but $$f\notin o(f)$$.