Are any of these functions of the same order?

Question

I ordered these functions by their asymptotic growth rates from smallest to largest.

$$\sqrt{2}^2, \log_{10}n, n^{1/2}, \log_2n^2, \log_2n^n,n^2, 2^n, (\log_2n)^n.$$

I am trying to figure out how to tell if any of them are in the same rate category? I read that if $\lim_{j\rightarrow\infty}\frac{f(n)}{g(n)}$ is a constant or infinity that they are the same order

Answer 1

The correct order would be (putting in the same line functions of the same order):

$$\sqrt 2^2$$

$$\log_{10}(n), \log_2(n^2)$$

$$\sqrt n$$

$$\log_2(n^n)$$

$$n^2$$

$$2^n$$

$$(\log_2(n))^n.$$