Finding argmin$_{n \in \mathbb{N}} |2^{n/12} - 5|$ non-computationally

Question

The problem is to find the integer $n$ such that $|2^{n/12} - 5|$ attains its minimum. Since it is clear that $24 \leq n \leq 35$, by computation one easily gets $n = 28$.

However, how to find this $n$ without any calculator?

Answer 1

I would try to solve the following equation:- $$ 2^{\frac{n}{12}}=5$$ by raising both sides to the $3rd$ power as the Right Hand Side (RHS) becomes quite close to an integer power of $2$ (fairly coarse approximation though) $$ (2^\frac{n}{12})^3=5^3 \Rightarrow 2^{\frac{n}{4}}=125\approx128 \Rightarrow n= 28$$