Proving negligible function

Question

I was reading the following:

The functions $2^{-n}, 2^{-\sqrt{n}}$ are negligible. However they approach zero at different rates. For example, we can look at the minimum value of $n$ for which each function is smaller than $\frac{1}{n^5}$

1. Solving $2^{-n} < n^{-5}$ we get $n>5\cdot log(n)$. The smallest integer value of $n>1$ for which this holds is $n=23$.

1- I don't understand why/how did they choose $1/n^5$ and not other function for comparison.

2- How to solve the inequality to be $n>5\cdot log(n)$ ?

3- How to find that the smallest integer is 23 without trying/guessing the numbers?

Answer 1

1. It's a more or less arbitrary example to illustrate a point.
2. \begin{align}&2^{-n} <n^{-5}\\ \iff &\log_22^{-n} < \log_2n^{-5}\tag{take $\log_2$}\\ \iff &-n < -5\cdot \log_2n\tag{use log rules}\\ \iff &n > 5\cdot \log_2n\tag{multiply by $-1$}\end{align}
3. It's a small number, so trying numbers starting at 2 should do the trick. You're checking that $\frac{n}{\log n} \leq 5$ and that function is monotone, so you could also do a binary search over an appropriate interval.