We now begin to see what the fractional answer that comes out of a logarithm actually means \(and why it is 316, not 500, which is the closest whole number to being 2\.5 digits long\)\. The logarithm base 10 of x is not answering "how many 10\-digits does it take to store x," it's answering "how many n\-digits does it take to store x, where all but one of those digits is a 10\-digit; and how big does the final digit have to be?"
What's n exactly?