Olympic Budget Passions

As you may know, even before the 2012 Olympics began, there was considerable discussion about the massive and burdensome budget for the United Kingdom. During the preparation for the Olympic Games, nearly every possibility for reducing expenses was considered.

One suggestion was to print athlete numbers using the binary numeral system, since printing the digits 0 and 1 on an athlete's number would cost only 1 shilling each, while printing any other digit would be more expensive. Although this proposal was ultimately not adopted and athletes received the standard decimal numbers (perhaps this is why the Olympics turned out so costly? :) ), your task is to determine how much it would have cost to produce an athlete's number in the proposed way if their number at the Olympics was $N$.