Question 2
Let $f(n)$ be the smallest number of $1$s needed to represent the positive integer $n$ using only $1$s, $+$ signs, $\times$ signs and brackets. For example, you could represent $80$ with $13$ $1$s as follows: $(1+1+1+1+1) \times (1+1+1+1) \times (1+1+1+1)$. Show that $3\log_3 n \le f(n) \le 5\log_3 n$ for $n \gt 1$.