I have a problem with definition of space constructable function.
As I understood we use this definition just for simplification of further proofs and idea behind this definition is very clear, but definition itself is very vague.
Sipser's definition: A function $f : \mathbb N \rightarrow \mathbb N$, where $f(n)$ is at least $O(\log n)$, is called space constructable if the function that maps the string $1^{n}$ to the binary representation of $f(n)$ is computable in space $O(f(n))$.
The role of this definition described very well in the book. But I just want to understand
Why we use boundary with $\log n$ function and not any other?
What's the role of $1^{n}$ input?
Thanks!