single-tape Turing machine

I am reading an introductory text on Turing machines( https://en.wikipedia.org/wiki/Turing_machine) and I have some questions. The first one is the following:
Prove that there is a single-tape Turing machine that doubles words of the form $1^n$ (result 1 to the 2n power), and works $O(nlog n)$ clock cycles.
Could you please help me to solve this problem?