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.