The Most Useful Curve in Mathematics [Logarithms]

Special thanks to the Patrons:
Juan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati

Welch Labs

References
The History of Mathematical Tables: From Sumer to Spreadsheets - Martin Campbell-Kelly
Navigation - James Pryde
e: the story of a number - Eli Maor
Description of the Wonderful Canon of Logarithms - John Napier
Construction of the Wonderful Canon of Logarithms - John Napier
The Daring Invention of Logarithm Tables - Klaus Truemper
A reconstruction of the tables of Briggs’ Arithmetica logarithmica - Denis Roegel
A reconstruction of the tables of Napier’s descriptio (1614) - Denis Roegel
The Polyphase Slide Rule A Self Teaching Manual - William E. Breckenridge
When Slide Rules Ruled - Cliff Stoll

[JonathanWaltersDrDub]

I'm almost convinced we should be giving students slide rules to teach them about logarithms. Sometimes touching the mathematics makes it more real. Thanks for your time in putting this together!

[AndrewDotsonvideos]

Really niche application warning : Logarithms (large ones) permeate so many theoretical nuclear physics calculations, especially ones describing processes where multiple, widely separated scales are relevant (eg. collider events where electron + positron --> 2 jets ). These large logs can ruin so many predictions in perturbative QCD if you're not careful. The expansion parameter (alpha_s) is small, but they multiply these large logs which ruins the convergence of the expansion. People then learned how to "resum" these large logs using things like renormalization group equations and effective field theories to obtain some of the most precise predictions in QCD to date (like extracting the value of alpha_s, the strong coupling constant). Logs almost ruined perturbation theory, but instead they suggested a more powerful way of predicting things perturbatively (N^kLL accuracy: Next-to^k Leading Log accuracy) in a lot of situations.

[adityakulkarni]

It also describes Welch Labs upload frequency 😢

PS Since calculators are banned upto high school, we still use log tables to do calculations during exams in India

[klausluger]

In honor of Henry Briggs I calculated logarithms of 10 from 1 to to 16 digits of precision, with following line of python np.log10(np.arange(1, int(1e6))), which instead of 7 years of my life, took around 7 ms of my life
I wonder how much the book cost in todays money when it was published. 7 years of mind melting labor must not have been cheap, so no wonder all the rest just copied his work for 300 years.
Then again if your work is used by next 300 years by literally EVERYONE you can be kind of proud of yourself

[eskay]

Before watching : 23 mins is really long
After watching : should be at least 2 hours

[SinanKaya-clho]

For anyone interested, the formula is: -10^7 * ln(x / 10^7)

(Napier's Logarithm)

[KipIngram]

16:34 - According to my calculator (a SwissMicros DM42 which does 34 digits of accuracy), Briggs's value for log(1.024) is correct out to the 952; after that he has a 6 and the calculator value has a 1. So 17 correct digits.

[douglasstrother]

The "Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables" by Milton Abramowitz & Irene A. Stegun contains a treasure trove of information, and not too expensive. Learning to read function tables is valuable in "sanity checking" hand and computer calculations.

[HiwasseeRiver]

I learned all three in school, logs, slide ruler, and cheap calculator - this was back in the 70's. In college we used another marvelous method - Nomographs. Layered onto that was dimensionless groups. You would be shocked how far that will take you in designing the modern world. Did I mention 3D models? We had them, they were physical models, but valuable tools all the same. We also had analog computers for heat transfer. You can use amps, ohms and volts to represent complex geometries. We also had a massive IBM computer and allocation of 1 second of computing time per semester.

[ZeDlinG]

In the first 60 seconds of the video you managed to show me WHY the log equivalencies are true, that my teachers failed for years.
I mean I now and use them, but I never SAW why they work, why multiplication becomes addition and so on
GOOD JOB!!!

[KipIngram]

I think that asking Napier to do that arduous task again was a bit much - I don't blame him for avoiding that.

[felipebarria]

Fascinating. I had never considered the origin of logarithms, I thought they had been defined simply to complete the triad of operations "Power-Root-Logarithm", but this is much more intuitive. There are certain things I had to pause and write down to get a good understanding, but I feel like I can try to teach this with a more open mind now.

[Samuirai]

I never understood these logic tables. Your explanation was so intuitive! Thank you!

[iteerrex]

I never ran across this amazing piece of history, but I did hear that we used slide rules to do all the science and engineering to go to the moon. Unbelievable! Thanks for a great video 👍

[ferenccseh]

In school, we were taught that "log<base a>(a^c) = c" meaning you can technically export the exponent of a number in a base. I found this explanation adequately useful (and I could remember the formula by saying AssAssin's Creed C [don't ask why that worked for me. Maybe bc it had the right number of letters in the right order?])

[samuelwaller]

This is an amazing video. I really appreciate that you went into depth on how they were actually calculated. The realization that you can essentially do a binary search with an iterative algorithm to find any value of a function is so, and even cooler when you learn that this is how computers calculate logarithms, trig functions, etc. to this day. Basically any time you can find a relationship where x/2 = f(y) or vice versa, you can do this. It is just so cool that you can do something a crazy as logarithms or trig *by hand* with enough will power, and it's not even that crazy difficult lol. I would love more content like this, so keep it up!

[Googahgee]

This is cool! Makes me realize that the term “logbook” is likely directly related to the logarithm, since it came from the “Ship’s Log.” I always used to think that logbooks (and related words) were just coincidentally the same as the word for logarithm, due to “logos” meaning knowledge and stuff, but it’s cool to see the connection between math and language!

[ben]

Thank you for taking me from vague idea of how those tables and slide rules worked to actual understanding. I'm thrilled to see anything you upload!

[MathHunter]

Love how you nailed saying a dozen times without messing up

[dwdei]

A beautifully put-together explanation. I like the touch of the basic animations.