How is $(2^a)^{\lg n} = n^a$?

Question

I was learning from Introduction to Algorithms (Chapter 3 under the topic “Logarithms”) and came across this expression.

$$ \lim_{ n \to 0 }{\frac{\lg^b n}{(2^a)^{\lg n}}} = \lim_{n \to 0} \frac{\lg^bn}{n^a} = 0 , $$ where $\lg$ is $\log_2$.

I don't understand how $(2^a)^{\lg n}$ changed to $n^a$. Please can someone explain it to me.

Answer 1

Because$$(2^a)^{\lg n}=2^{a\lg(n)}=\left(2^{\lg(n)}\right)^a=n^a.$$