Introductory guide to logarithms

https://arbital.com/p/log_guide

by Nate Soares May 27 2016 updated Oct 21 2016


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:

$$~$\underbrace{139}_\text{3 digits}$~$$

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.

%box: This path contains 9 pages:

  1. What is a logarithm?
  2. Log as generalized length
  3. Exchange rates between digits
  4. Fractional digits
  5. Log as the change in the cost of communicating
  6. The characteristic of the logarithm
  7. The log lattice
  8. Life in logspace
  9. The End (of the basic log tutorial) %

Comments

Eric Rogstad

It's OK if this doesn't make sense yet; that's what the tutorial is for\. It's also OK if this interpretation raises more questions than it answers: Why are the logarithms above using base 10? What does it mean to say that a number is "two and a half digits long"? Why is it 316 \(rather than 500\) that is using two and a half digits?

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…"

Eric Rogstad

Welcome to the Arbital introduction to logarithms\! In modern high schools, logarithms are commonly mentioned and rarely motivated\. At best, young students are told that logarithms are merely a tool for inverting exponentials\. At worst, they're simply told a bunch of properties of the logarithm that they're supposed to memorize\. The goal of this series is to explore what logarithms are actually doing, and help you build an intuition for how they work\.

I'd take out 'young'

Eric Rogstad

I did it.

Eric Rogstad

You might also be wondering what it would mean 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 2\.5 digits long — 500 is using nearly 2\.7 digits\. Why? What does that mean?

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