Is it true that $n/\log(n)$ is approximately $\log(n/\log(n))$ for large enough $n$

Question

Someone wrote this on a homework assignment I'm grading:

$n/\log(n)$ is approximately $\log(n/\log(n))$ for large enough $n$

Is there an easy way to see it as true or false?

Answer 1

No since $\frac{\frac{n}{log{n}}}{\log{\frac{n}{log n}}}\rightarrow \infty$

Answer 2

We can just define $m=\frac n{\log n}$. This function is monotonic when $n$ is large enough. Then the claim is that $m \approx \log m$, which is badly wrong.

Answer 3

No, it's like asking if t is approximately log(t) when t is really big, which is like asking if p is approximately 2^p when p is really big, and that's obviously wrong.