"I think you may need to spe..."

https://arbital.com/p/425

by Eric Rogstad Jun 7 2016


Specifically, if a number is digits long \(in decimal\_notation\), then its logarithm \(base 10\) is between and \. This follows directly from the definition of the logarithm: is the number of times you have to multiply 1 by 10 to get and each new digit lets you write down ten times as many numbers\. Thus, the number of digits you need to write is close to the number of times you have to multiply 1 by 10 to get \. 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.