Does the logarithm function grow slower than any polynomial?

Does $f(x)=ax^b$ grow faster than $g(x)=\ln x$ for all $a, b > 0$? Can I say that $f(x) > g(x)$ as $x$ approaches infinity?

I thought the answer is yes, but this graph appears to be telling a different story.

Is the polynomial (the green curve) going to cross the log function (the red curve) and exceed in value for some large value of x?

If the answer is yes, does it mean that if I subtract the two functions and set it to zero, the resulting equation will have two roots? What are the roots?

Solution 1:

The answer is yes, although in some cases (like the one you have given) it takes a very long time for the polynomial function to catch up to and ultimately dominate the log function.

A rigorous formation of what you are saying is: $$ \lim_{x \to \infty} \frac{\log(x)}{P(x)}=0$$

where $P(x)$ is any polynomial. The limit tending to zero just means that the bottom terms dominates as $x \to \infty$.

Here is a proof of the limit equality for the case of $P(x)=x^b$ for some $b>0$. The case of polynomials follows as an easy corollary.

$$ \lim_{x \to \infty} \frac{\log(x)}{x^b} = \lim_{x \to \infty} \frac{1/x}{bx^{b-1}} = \lim_{x \to \infty} \frac{1}{bx^b}=0$$

where the first equality follows from l'Hôpital's rule.