Which one is bigger for large value of $n$?

Question

I stumbled upon this question, among $n^{\frac{7}{4}}$ and $n(\log^9(n))$, which one is bigger?

I think that leaving out n from both leaves $n^{\frac{3}{4}}$ and $\log^9(n)$.

Now let's take $n^{\frac{3}{4}} = n$.

So, according to the concept of exponential, they grow very fast making $\log(n)$ very small. Thus I can conclude that multiplying $9$ parts of this $\log(n)$ we can never make it equal to $n$.

This is what I can imagine. I don't know even if it is correct or not. Can anyone please say if I'm on the right track? Also if you can say how to deduce the conclusion will be great. Thanks in advance.

Answer 1

The quick fact to remember is that any power function (with a positive exponent) always wins over any power of logarithms.

We can compare $$n^{\varepsilon} \lessgtr \log^a n$$ by taking the $a$th root on both sides which gives $$ n^{\varepsilon/a} \lessgtr \log n = \frac{1}{\varepsilon/a}\log(n^{\varepsilon/a}) $$ Since $n^{\varepsilon/a}\to \infty$ is the same limit as $n\to\infty$, once we know that $\log n = o(n)$, this automatically generalizes to positive powers on both sides.

Answer 2

Apply l'Hopital's rule (repeatedly) to the limit $$\lim_{n\to\infty}\frac{n^{\frac34}}{\log^9(n)}.$$ What does this tell you?

Alternatively, set $x:=\log n$ and substitute this to get the functions $$n^{\frac34}=(e^x)^{\frac34}=e^{\frac34x}\qquad\text{ and }\qquad \log^9(n)=x^9.$$ Can you tell which one grows faster then?

Answer 3

With asymptotic analysis: $\ln^an=o(n^b)$ near $\infty$ for all $a,b>0$. So $$n\ln^9n=n\,o\Bigl(n^{3/4}\bigr)=o\Bigl(n^{1+3/4}\bigr).$$