Specifically, if a number $~$x$~$ is $~$n$~$ digits long \(in decimal\_notation\), then its logarithm \(base 10\) is between $~$n-1$~$ and $~$n$~$\. This follows directly from the definition of the logarithm: $~$\\log\_{10}(x)$~$ is the number of times you have to multiply 1 by 10 to get $~$x;$~$ and each new digit lets you write down ten times as many numbers\. Thus, the number of digits you need to write $~$x$~$ is close to the number of times you have to multiply 1 by 10 to get $~$x$~$\. The only difference is that, when using digits to write numbers down, you only get to use whole digits, whereas when computing logs, you can multiply 1 by 10 fractionally\-many times\.

I think you may need to spell out this 10 times as many numbers part. This is a large unexplained step in explaining why the log is the length.