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?