Consider the following functions, where $n$ is a positive integer. $$n^{1/3}\ ,\ \log(n)\ ,\ \log(n!)\ ,\ 2^{\log(n)}$$ Which one of the following options lists the functions in increasing order of asymptotic growth rate? *Note:* Assume the base of $\log$ to be 2.

To compare the asymptotic growth rates of the given functions, let us simplify and analyze each one as $n \to \infty$:

1. $\log(n)$: A logarithmic function, which grows slower than any polynomial function $n^c$ for $c > 0$.

2. $n^{1/3}$: A polynomial function with degree $\frac{1}{3}$. Since $\lim_{n\to\infty} \frac{\log n}{n^{1/3}} = 0$, we have $\log(n) = o(n^{1/3})$.

3. $2^{\log_2(n)}$: Using the identity $b^{\log_b x} = x$, this simplifies directly to $n$. Since $\lim_{n\to\infty} \frac{n^{1/3}}{n} = 0$, we have $n^{1/3} = o(n)$.

4. $\log(n!)$: By Stirling's approximation, $\log(n!) = \Theta(n \log n)$. Since $\lim_{n\to\infty} \frac{n}{n \log n} = 0$, we have $n = o(n \log n)$.

Arranging the functions in increasing order of asymptotic growth rate gives:

This matches Option (A).