# 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:

Parents:

• Logarithm
• Mathematics

Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.

• 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…”

• I’d take out ‘young’

• I did it.

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