# Do exponential functions grow faster than logarithmic?

For example:

$$f(n) = n(log_2 n)^4$$ vs $$g(n) = n^{1/3}$$

Does f(n) grow slower than g(n) as a general rule, even for decimal exponents? I tried doing a limit test but symbolab limit calculator is saying that Steps are currently not supported for this problem.

I graphed them over a large range and it does appear that g(n) grows asymptotically faster.

I think you may be confused about the definition of Big O - graphing f(n) and g(n) shows f(n) surpassing g(n) at nearly every scale.
O(f(n)) > O(g(n) because g(n) is a fractional power polynomial, which is by definition sublinear (if you linearly plug numbers into a square root, cube root, etc., the rate at which the output increases decreases with each successive computation). g(n) is a "n log star n" polynomial meaning it follows the form of a log to a power multiplied by n. The rate of growth of these types of functions far exceeds fractional polynomials and its not hard to see why - n is a term of the function, so we know its at least linear.