The End (of the basic log tutorial)

That con­cludes our in­tro­duc­tory tu­to­rial on log­a­r­ithms! You have made it to the end.

Through­out this tu­to­rial, we saw that the log­a­r­ithm base \(b\) of \(x\) calcu­lates the num­ber of \(b\)-fac­tors in \(x.\) Hope­fully, this claim now means more to you than it once did. We’ve seen a num­ber of differ­ent ways of in­ter­pret­ing what log­a­r­ithms are do­ing, in­clud­ing:

For ex­am­ple, \(\log_2(100)\) counts the num­ber of dou­blings that con­sti­tute a fac­tor-of-100 in­crease. (The an­swer is more than 6 dou­blings, but slightly less than 7 dou­blings).

We’ve also seen that any func­tion \(f\) whose out­put grows by a con­stant (that de­pends on \(y\)) ev­ery time its in­put grows by a fac­tor of \(y\) is very likely a log­a­r­ithm func­tion, and that, in essence, there is only one log­a­r­ithm func­tion.

We’ve glanced at the un­der­ly­ing struc­ture that all log­a­r­ithm func­tions tap into, and we’ve briefly dis­cussed what makes work­ing with log­a­r­ithms so dang use­ful.

There are also a huge num­ber of ques­tions about, ap­pli­ca­tions for, and ex­ten­sions of the log­a­r­ithm that we didn’t ex­plore. Those in­clude, but are not limited to:

  • Why is \(e\) the nat­u­ral base of the log­a­r­ithm?

  • What is up with the link be­tween log­a­r­ithms, ex­po­nen­tials, and roots?

  • What is the deriva­tive of \(\log_b(x)\) and why is it pro­por­tional to \(\frac{1}{x}\)?

  • How can log­a­r­ithms be effi­ciently calcu­lated?

  • What hap­pens when we ex­tend log­a­r­ithms to com­plex num­bers, and why is the re­sult a mul­ti­func­tion?

An­swer­ing these ques­tions will re­quire an ad­vanced tu­to­rial on log­a­r­ithms. Such a thing does not ex­ist yet, but you can help make it hap­pen.