5. Given the sequence $\left\{a_{n}\right\}$, where $a_{1}=2, a_{n+1}=\left(a_{n}\right)^{2}\left(n \geqslant 1, n \in \mathbf{N}_{+}\right)$, which of the following conclusions is correct? ( ).
A. $a_{n}$ is an arithmetic sequence
B. $a_{n}$ is a geometric sequence
C. $\lg a_{n}$ is an arithmetic sequence
D. $\lg a_{n}$ is a geometric sequence
5. D. Taking the logarithm of both sides of $a_{n+1}=\left(a_{n}\right)^{2}$, we have $\lg a_{n+1}=2 \lg a_{n}$, so $\left\{\lg a_{n}\right\}$ is a geometric sequence.