Proving log(n)^3 = O(n^(1/3)) using Big O notation

[hosman]

is log(n)^3 O(n^(1/3))??

Homework Statement

So the problem has to do with big o notation, I came up with a solution but I think even in this summed up solution I give I am making too much assumptions or that it is just plainly wrong, if it is please let me know.

Homework Equations

So the original question is stated as follows:

Q:Show that $$\log{n}^{3}=O(n^{1/3})$$

The Attempt at a Solution

My basic idea was to use the derivative of the third root of both functions and prove that $$n^{1/9}$$ will have the greater one of the two and thus at some point the two functions will be equal and then $$n^{1/9}$$ will become greater allowing $$\log{n}$$ to have a order complexity of $$n^{1/9}$$,
which implies $$\log{n}^{3}=O(n^{1/3})$$

so here it is:
if $$n^{1/3}$$ has $$O(n^{1/3})$$ the so should...
$$\sqrt[3]{n^{1/3}}$$ have $$O\left(\sqrt[3]{n^{1/3}}\right)$$
and so I take the derivative of the third root (to simplify problem) of these functions and get:

$$\frac{d}{dn}\left(\log{n}\right)=\frac{1}{n\ln{10}}$$

$$\frac{d}{dn}\left(n^{1/9}\right)=\frac{1}{9n^{8/9}}$$

clearly $$\frac{1}{n}<\frac{1}{n^{8/9}}$$ for big values of n.

thus the derivative of $$n^{1/9}$$ will eventually become greater than $$\log{n}}$$ and this means (leaving out the actual calculation of following constants):

$$\exists N,c\in\Re$$ such that $$\forall n \geq N$$

$$\log{n} \leq c\left(n^{1/9}\right)$$ so that (skipping some more steps..)

$$\log{n}^{3}=O(n^{1/3})$$

for the sake of brevity I am leaving out steps but I think that the solution still makes sense?

 

[hosman]

I just found out that the actual problem is litte o not big O so its
Q: proof (logn)^3 is o(n^(1/3))

hmmmmm...

 

[Mene]

Your solution is mostly correct, but there are a few things to clarify. First, in order to prove that log(n)^3 = O(n^(1/3)), we need to show that there exists a constant c such that for all n greater than some value N, log(n)^3 is less than or equal to c*n^(1/3). You have correctly shown that the derivative of n^(1/9) eventually becomes greater than the derivative of log(n), but this does not directly prove that log(n)^3 is O(n^(1/3)).

To prove it, we can use the fact that the derivative of n^(1/9) is always positive, while the derivative of log(n) is only positive for n greater than 1. Therefore, we can choose a value for N such that for all n greater than N, the derivative of n^(1/9) is greater than the derivative of log(n). This means that for all n greater than N, n^(1/9) will eventually become greater than log(n), and thus log(n)^3 will also eventually become less than or equal to n^(1/3).

So, by choosing a constant c that is greater than the value of n^(1/9) at N (since n^(1/9) is positive), we can show that log(n)^3 is indeed O(n^(1/3)).

In summary, your solution is on the right track, but it is important to clarify the steps and assumptions in order to fully prove the statement using Big O notation.