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:

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

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) <div>