# Introductory guide to logarithms

Welcome to the Arbital introduction to logarithms! In modern education, logarithms are often mentioned but rarely motivated. At best, students are told that logarithms are just a tool for inverting exponentials. At worst, they’re told a bunch of properties of the logarithm that they’re expected to memorize, just because. The goal of this tutorial is to explore what logarithms are actually doing, and help you build an intuition for how they work.

For example, one motivation we will explore is this: Logarithms measure how long a number is when you write it down, for a generalized notion of “length” that allows fractional lengths. The number 139 is three digits long:

and the logarithm (base 10) of 139 is pretty close to 3. It’s actually closer to 2 than it is to 3, because 139 is closer to the largest 2-digit number than it is to the largest 3-digit number. Specifically, \(\log_{10}(139) \approx 2.14\). We can interpret this as saying “139 is three digits long in decimal notation, but it’s not really using its third digit to the fullest extent.”

You might be thinking “Wait, what do you mean it’s closer to 2 digits than it is to 3? It plainly takes three digits: ‘1’, ‘3’, and ‘9’. What does it mean to say that 139 is ‘almost’ a 2-digit number?”

You might also be wondering what it means to say that a number is “two and a half digits long,” and you might be surprised that it is 316 (rather than 500) that is most naturally seen as 2.5 digits long. Why? What does that mean?

These questions and others will be answered throughout the tutorial, as we explore what logarithms actually do.

Parents:

Instead of telling me “It’s OK if this doesn’t make sense…” (No it’s not okay! I was told this is a site where things make sense!) say a sentence that I find myself agreeing with, like:

“Wait, what do I mean that 139 is “close to” being a three digit number!? There are plainly three digits, ‘1’, ‘3’, ‘9’, sitting right there on the page. For the answer, read on…”

Fixed.

I’d take out ‘young’

I did it.

I would just take “500 is using nearly 2.7 digits” out.