The Most Useful Curve in Mathematics [Logarithms]

[@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.

[@hectorbetancourt2854]

Hey Andrew! So nice to see you here, it's been a while since I saw one of your videos, but you, along with Zach Star, were one of my "gateway" science communicators all those years ago. I am now beginning my Masters in Advanced Manufacturing Systems, and I wanted to thank you for being an inspiration. Hope to see more sketches on your channel once you are done with your PhD. Cheers!

[@oni8337]

holy shit its andrew dotson

[@scheimong]

Damn. Is this English I'm reading?

[@WelchLabsVideo]

Woah interesting

[@SorobanWorld]

I don't know if this will help you at your very high level of mathematics, but ... another valid form of logarithm is not a decimal number but a continued fraction. Viewing logarithms as decimals seems like a necessity, but it is not the only form that they can take. There is an abbreviated form of continued fraction notation that I like at my hobbyist level: the fraction 1/3 can be represented by [0;3]. My general point is that the paradigm of logarithms only as decimals might cause problems. Even simple arithmetic with continued fractions is its own issue, but the idea of paradigm paralysis is still something, you know? The representation of numbers colors the perspective of the math involved.

[@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!

[@colinunsworth8822]

I was an engineering student in the late 60's, and although I used a sliderule exclusively and knew the reason why it worked, I really didn't think much about it. As long as I kept track of where the decimal place was!

[@petercalkins3011]

@@colinunsworth8822 : And that is the beauty of the slide rule - one can appreciate the movement of the decimal point. This fact is lost on calculator users.

[@ivoryas1696]

​@@colinunsworth8822 Honestly... _still_ sounds preferable to the current day...

[@derrellrose7020]

I think that would help me. This video helps. HS did not.

[@tongpoo8985]

Yeah good point. When I was a kid in elementary school, this is how we were taught math. They'd give us all kinds of doohickeys to help visualize stuff and play with. Helps you wrap your mind around things, you just enjoy playing with the thing and your brain makes all sorts of connections naturally. Like they'd give us these little cubes that each weighed a gram and have us do comparative measurements on one of those lever scales with it. Then they'd give us this puzzle where a bunch (27) of these cubes were attached in a certain way by a string, and to solve the puzzle you have to fenagle it to get it as a 3x3x3 cube. Then you get one of those lever scales and see that it's balanced between the puzzle cube and 27 individual ones. Then you count all the cubes in the puzzle and see that it's 27. It's kind of a random-seeming number so you think about it. You notice that each slice of the cube is 9 cubes, which you already know is 3x3. And you notice that there are 3 slices, and 3 times 9 is 27. All they did was give us the cubes, the scales, and the puzzle, and show us that the puzzle can be made into a cube, yet we developed intuition for the fundamentals of exponentiation, volume and area, how volume relates to mass, and a lot of spatial reasoning from solving the puzzle, just from playing around with it on our own and without even knowing what exponentiation is yet. Was a really good elementary school imo. They would also give us daily verbal logic puzzles.

[@eskay1891]

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

[@WelchLabsVideo]

Yeah I originally thought this was going to be way shorter - but it got kinda deep!

[@GPP_feature42]

​@@WelchLabsVideoI really enjoyed the presentation and pacing. Your team all did an excellent job, many thanks 👍🌞

[@wofite5989]

Ima be honest I thought I already replied because of my pfp

[@bbhrdzaz]

I thought so too, but the argument is compelling

[@ouroboros7388]

Me reading the comments before watching the video, to see if would be worth it over Idk, another video 😅: Yea, must be exaggerating, but I'll give it a go nonetheless Me after watching the video: when is the missing hour and 37 minutes gonna be released?

[@klausluger7671]

In honor of Henry Briggs I calculated logarithms of 10 from 1 to 10000000 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

[@altrag]

> I wonder how much the book cost in todays money when it was published Probably not much - at least not significantly more than any other book at the time. It was a couple hundred years before the invention of capitalism and a lot of this kind of work was done by rich people just because they were interested in doing it. The primary "currency" they were looking for was reputation, not physical wealth. You often self-published either on your own dime or that of a patron just enough copies to send to those who you thought might be interested (or that you wanted to brag to). The printing press made doing so a heck of a lot cheaper and easier to be sure, but it was still nowhere near comparable to the millions of copies sent all over the world for consumption by the general public that we see today. That's not to say books weren't bought and sold - they absolutely were - but mostly as a secondary market. Sold off because the owner needed money or died and their inheritors didn't care about books or straight up stolen/looted by thieves. (At least for this kind of works. Things with regular editions and broad audiences like almanacs and trade pamphlets are a different story of course - those were much more widely published in a manner similar to today's publishing industry.)

[@joansparky4439]

​ @ altrag books were sold for as much as it cost to make (and copy) them. If it wasn't sustainable, the author did it once and that was it. After the printing press was invented this process became cheaper and publishers paid authors for first print exclusivity, but after that everybody had a go at it - the author got it's share and the publisher had to see how to make the revenue work for themselves to sustain it (books were cheap and plenty)... and then copyright got invented in the UK in the 17th century and the publishers were able to control the supply AGAINST CAPITALIST PRINCIPLES by disabling competition. One of the reasons Germany was able to catch up to the UK was because copyright got introduced there only a century later - so books that distributed knowledge and information were plenty and cheap in Germany at that time.

[@MaximumBan]

He used a computer. From wiki: According to the Oxford English Dictionary, the first known use of computer was in a 1613 book called The Yong Mans Gleanings by the English writer Richard Brathwait: "I haue [sic] read the truest computer of Times, and the best Arithmetician that euer [sic] breathed, and he reduceth thy dayes into a short number." This usage of the term referred to a human computer, a person who carried out calculations or computations. The word continued with the same meaning until the middle of the 20th century. During the latter part of this period women were often hired as computers because they could be paid less than their male counterparts.[1] By 1943, most human computers were women.[2]

[@shardinalwind7696]

You only calculated from 1 to 1000000, not 10000000. You should have typed in 1e7 for the result you wanted

[@mike74h]

​@@shardinalwind7696 Maybe he meant 1000000.

[@adityakulkarni4549]

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

[@andersjjensen]

That is dumb as rocks. There is absolutely no difference punching in the numbers and getting the answer... and looking up the answer in a book.

[@ArmedVapor]

​@@andersjjensenyou are correct there is no difference. This method just isn't for lazy people. Believe it or not some enjoy the puzzle aspect of a problem.

[@hastypete2]

@@andersjjensen Agreed for using tables. and yet... Knowing how to use a slide rule really does enhance understanding. Maybe they should require slide rules and ban tables too! 😄

[@Archiekunst]

@@andersjjensen speaks a lot about the indian education system. What is rote is praised.

[@hanifarroisimukhlis5989]

@@andersjjensen Not really, it teaches you how these systems works. Ancient, yes. But still works and useful. Also big note here, cheating with rigged calculator is *very easy* to do.

[@SinanKaya-cl5ho]

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

[@opensocietyenjoyer]

it's just a log

[@SinanKaya-cl5ho]

@@opensocietyenjoyer just using the log does no give 28804057 from 561000

[@crix_h3eadshotgg992]

Am I retarded or does that simplify to ln(x)?

[@klasta2167]

remove negative

[@_Hawken]

​@@SinanKaya-cl5hoit will be the right way around then Checked on Desmos

[@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.

[@movin3148]

I’d love a set of slide rules that had saturation data, or even if there was a way to add PVT gas properties onto slide rules I’d buy them any day. It’s crazy how we still have to use NIST tables or have to rely on software calculators for quick hand calcs, it’s not quick at all

[@KipIngram]

I heard the phrase "dimensional analysis" when I was in college, in the 80's, but it was little more than "always check your units." I didn't discover until decades later, just because of prowling around the internet endlessly, the real full scope of that topic. I feel sure that in earlier times it was taught as a routine part of an engineering education - I really hate it that some of those great ideas have fallen off the radar.

[@ViKODiN_]

Just out of curiosity, what would be reasonable and practical to compute with one second of time on the IBM computer? Surely you could only get the results of one equation or so, so you had to be sure that was the one you wanted 😂

[@Grateful92]

​@@KipIngramwould you like to share the url link of that source from which you learned dimensional analysis, it would help me too and I will be thankful to you for sharing it 😊

[@ferenccseh4037]

In school, we were taught that "log(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?])

[@thehemperor3967]

I have a few really silly mnemonics for math too. F.e.: I always remember the trig functions with an association to Lady Gaga (That works only in German tho). To remember surjectivity and injectivity of graphs, I always say to myself: "A positive parabola eats a Surschnitzel (that's a special kind of schnitzel)", because when f(x) = x², where R->R+ (cutting of negative y axis) the graph is surjective, but not injective. And for the roots of a complex number, I made myself the pikachu rule. Adding 2*pi*k to theta (polar angle), were k goes from 0 to n-1 (where n describes the n-ths root, because there are n roots of the n-th root of a complex number), before dividing by n, gives all roots. 2*pi*k = pi*k + pi*k (which sounds like pika pika)

[@SirNobleIZH]

I first learned log as what exactly it did as a function directly, that being, if a^b=c, then log (c) = b Built my intuition from there. Like how learning what exponents do, then the other things they're used for I learned in that order so the power extraction is for me the most intuitive part

[@douglasstrother6584]

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.

[@shasan2393]

Thank you for recommending this

[@ZeDlinG67]

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!!!

[@samuelwaller4924]

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!

[@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 👍

[@isaacwolford]

Well… that’s partially true. Computers were also heavily used. Here is a summery from Google: During the Apollo missions, an on-board computer and large computers on Earth performed the critical guidance and navigation calculations necessary for a successful journey. In addition, crews carried a slide rule for more routine calculations.

[@WelchLabsVideo]

Thanks for watching!

[@richardnineteenfortyone7542]

A lot of work was done on slide rules in the 1960's only because managers sequestered computers for "data processing". That meant payroll and accounting. And engineers were not allowed access because they might acquire marketable skills. Barney Oliver began the climb out of this darkness. But his HP-35 was still worthless for optical ray tracing because the sines of very small angles were not accurate. This was fixed in HP-45. Every time you hit that button, you save the time of looking in a table and doing the interpolation! And you avoid the chance of error in calculation or copying the results from one paper to another. At one airplane factory where I worked, certain programmable calculators were declared "unauthorized data processing instruments" and were banned (until the program collapsed and the entire division was dissolved in 1993). But we all used them anyway, knowing how such rules originated. By this time we were starting to get '286 machines, but were not allowed any meaningful engineering software, nor could we get ink ribbons for our printers. A little WD-40 could give a ribbon extended life, but eventually...

[@firstname4337]

LOL @ believing we actually went to the moon

[@Grizzly01-vr4pn]

@@firstname4337 Is that the royal 'we'?

[@felipebarria3204]

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.

[@vibaj16]

I just realized that the reason there's these 3 operations related to exponents (power/root/log) instead of 2 operations like with addition (add/subtract) and multiplication (multiply/divide) is that addition and multiplication are commutative, unlike exponentiation.

[@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.

[@calyodelphi124]

This video makes me feel so happy, because starting a couple of years ago I seriously got into slide rules because they are just so wonderfully hands on. I now have two circular slide rules and one old (and rudimentary) linear slide rule in my collection, with aspirations to collect more.

[@Samuirai]

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

[@defragsbin]

Nice video. It frustrates me that the history & practical use of mathematics is often an afterthought. I learned and used logarithms at high school & university, but to me it's just an abstract thing. If you ask a random person in the street what a logarithm is (even someone who has learned & used them in education), they will likely shrug. To me, education should start with something like this video! It's really motivating to understand how these techniques revolutionised travel & other parts of life.

[@ben9089]

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!

[@leexcite2903]

Instant subscribe. U break down concepts so easily that I start to appreciate them now

[@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!

[@CallOfCutie69]

wow

[@CallOfCutie69]

But they say that ship’s log comes from chip log, an instrument

[@johncherry108]

I've only ever heard "logbook" used to refer to a record of service for a car, or some other form of transport. When I finished high school and commenced Engineering at university in the seventies, we used books of logarithms because I couldn't afford the fancy new HP35 calculator. I used a simple slide rule so the connection between logarithms and slide rules has always been obvious to me. Nowadays, logarithms and slide rules are historical relics. I still enjoyed Veritasium's video.

[@ChrisTaylor-NEP]

@@CallOfCutie69 Can confirm. The log literally was a log that you threw into the water so that the ship's speed could be calculated from how fast it passed the log. This figure was then recorded in the ship's log. Interestingly, when I became a navigator some 4 decades ago, using logarithms was considered the only reliable way to do celestial navigation and ship stability calculations. Even after satellite navigators were installed on board, we still used sextants and logarithms to check that the satellite navigator was displaying the correct position.

[@CallOfCutie69]

@@ChrisTaylor-NEP must have been an exiting job. It’s interesting how you can find people with careers spanning so long in KZbin comments. I hope you’re doing well. I myself would be afraid of any position that exposes me to the sun, because of photo damage to the skin. Do you feel being outside so much ages you faster?

[@v8pilot]

As a radio enthusiast from the age of 11 (made my first crystal set then with one of the new fangled germanium diodes, price 2/6), I cannot remember not understanding logarithms. It helped that my dad was a Cambridge educated engineer.

[@jeffdo1974]

I loved how you tied it back to early tech. It reminds me of my dear old dad's slide rule and calculator. As a kid as I fascinated with both. Great to learn some history behind it! Thanks

[@WelchLabsVideo]

Glad you enjoyed it!

[@dwdei8815]

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

[@felixsteinbeck2878]

@Welch Labs: At Minute 13:39, shouldn't it be y/2 for Sqrt(x) ? Your example is: x = 10 --> y = 1 Let's call x2 = Sqrt(x) --> y2 = 1/2 ==> y2 is 0.5 * y, not 0.5* x = 10. Or where is my misunderstanding coming from? Given y is log 10 (base 10), and at minute 13:45 you even write "...= 1/2 * log 10 (base 10) = 1/2 "

[@matheusstauffer7235]

Watching this video was like contemplating a work of art. Math is wonderful. Great work, Welch Labs team!

[@Piokoxer]

Logarithms are super useful in game design making something scale infinitely but also slow down as you go is really easy with logarithms and super useful That's a practical application for ya

[@MathHunter]

Love how you nailed saying 9999999 a dozen times without messing up

[@PabloMoscato]

Congratulations! I added to a playlist called "Computing Fundamentals". I think that videos like this contribute immensely to give a historical framework to the formulas and the math and is formative and influential to many students. They contribute to show next generations the incredible efforts of these pioneers in mathematics (and difficult computations). Great example! Thanks for contributing this that benefits the community at large.

[@CL2K]

Looks like it turned division from an O(n^2) algorithm into an O(n) algorithm!

[@FutureAIDev2015]

Last Friday my calculus professor introduced the class to logarithmic differentiation, and it makes solving the problems so simple that I have a feeling I'm not going to be using the quotient rule anytime soon

[@nateradetunes]

Very cool to learn that slide rules rely on logarithms, I didn't realize that. I knew about log tables but I didn't realize that the slide rule itself was an embodiment of this "easier calculation" quality of logarithms. :)

[@martincohen8991]

From the functional equation for logs, f(xy)=f(x)+f(y), you can show that its derivative f'(x)=f'(1)/x, so, since f(1)=0, f(x)=integral from 1 to x of f'(1)/t dt. The natural log of x, to base e, is gotten by chosing f'(1)=1. From this you can get the power series valid for -1

[@AuroraNora3]

Circular slide rules are still used by student pilots in training. The circular construction of the slide rule enables additional functionality related to temperature, wind, speed, pressure-altitude, etc. See: The CRP-5

[@brockobama257]

"You can't outdo me, I'm the god of rhythm All natural like the LOGARITHM" - 3Blew1Blown by JoFo

[@lanimulrepus]

Amazing video... I will look at the stack of slide rules buried in dust on my bookcase shelves, which got me through college, grad school, and the early part of my career, with a new appreciation.

[@ivoryas1696]

Man I *_was_* just gonna watch the intro and then watch the rest later but, logarithms man... It's like one of the _progenitors_ of data compression and that's just fascinating even though I mostly just study CS and pure mathematics on the side, lol.

[@TheJara123]

The level of detail and critical approach in your videos unparreled man!! Deeply impresed!! Waiting for your book.

[@ericispublius]

An incredible video that breaks them down and helps understand what they are! I wish that I had seen this before I went to college. I have an animation question, if you have a moment! How did you animate the numbers come off of the page at 5:42? I'm working on some projects to use my physical typefaces, and I think that would be so handy!

[@WelchLabsVideo]

Thanks for watching! I just used illustrator & premiere.

[@davidcottrell1308]

Thank you...Brilliant job! Great and exemplary use of music. Just frickin' fantastic.

[@douglasstrother6584]

My venerable HP32S that I got in 1989 really *is* an electronic slide rule. That was a fun video!

[@richardnineteenfortyone7542]

Time to move up. My current favorite is the WP-34s. It is built on the HP-30 business analyst. But for real power, get the free iPhone app or the PC emulator; runs a hundred times faster on long programs. The iPhone app has keys that won't go bad like the HP post Fiorina hardware. HP Prime is also available as an iPhone and a PC app. But it has nothing to do with HP; as far as I know it is a development of the US Royal Typewriter company. There is more inside that beast than you can ever learn.

[@raymitchell9736]

A very nice piece of history and mathematics wrapped together. Of course you know about Charles Babbage's Difference engine and how it was proposed to be used to solve polynomials. I got to see one in action at the History of Computing, Paul Alan (of Microsoft fame) commissioned it to be made with intentional machining errors to replicate the technology of the day when Babbage was alive, just to see if it were built, would it work. And the answer was yes, but the crank mechanism would have had to be beefed up, but still in the realm of what could be possible. It was a joy to watch, and they loaded the cams as the input of the polynomial to be solved and set the wheels, then started cranking out the answers. They had a mechanical printer and... oh my gosh you have to find out more about it, it's more than I can share in a comment. I think there's more about it online. Cheers!

[@gregwochlik9233]

I am a proud owner of 3 slide rules. When I did my attempt at pilots license in 2007 ~ 2009, We had to purchase and use a circular vesion of the slide rule. It is called a "E6-B Flight computer". Mine has survirved.

[@richardnineteenfortyone7542]

E6-B? I used my father's from the 1950's. But it lacked a scale for density altitude. I worked out a simple way to do DA, but the flight instructor went ballistic, "You can't do that!" It gave the right answers, what's he complaining about? Not flying any more due to glaucoma.

[@siddharth_desai]

The fact that "there exists a group homomorphism from the reals under addition to the +ve reals under multiplication" still feels like magic to me.

[@WelchLabsVideo]

I know right! It's like another dimension.

[@cken27]

Amazing history and amazing explanation. Thank you for your hard work in production of this video!

[@WelchLabsVideo]

Thanks for watching!

[@rebase]

Interestingly, there exist other functions that can convert multiplication into addition/subtraction. E.g. consider the function F(x) = x^2 / 2 Then for any a and b: F(a) = a^2 / 2 F(b) = b^2 / 2 F(a - b) = (a - b)^2 / 2 = a^2 / 2 - ab + b^2 / 2 = F(a) + F(b) - ab Thus: F(a) + F(b) - F(a - b) = ab So you convert multiplication into two subtractions and two additions.

[@ossigaming8413]

One rathee want: f(ab) = f(a) + f(b)

[@rebase]

@@ossigaming8413 it depends. To calculate a product with F(ab) = F(a) + F(b) one has to do two lookups, one addition and an inverse lookup. With the half-square function you don't need the inverse.

[@richardnineteenfortyone7542]

Try cosα*cosβ=[cos(α−β)+cos(α+β)]/2. Tables of half cosines were published to facilitate this method.

[@sanveersingh3513]

missed your explanations. Appreciate your great work

[@GopnikStar]

It bothered me when I asked how logs work to my maths teacher and just got a response like "It's just how you figure this out". This is super helpful! but I can't even remember the types of questions we were using them on 😅

[@KipIngram]

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

[@richardnineteenfortyone7542]

One other comment: The advent of logarithms greatly facilitated the reduction of astronomical data. I don't know if Galileo used logs, but his false teachings about the sun being the center of the universe got him into bad trouble with the church. The real devil was Gutenberg, whose printing press made heretical ideas widely available. And about this time baroque music started and the American civilizations in Mexico and Peru were conquered. A time of very rapid change. And, of course, those heroic pilgrims landed in America seeking religious freedom. That meant the freedom to expel from the community, imprison, torture, or put to death anybody who does not rigidly obey their clergy.

[@wilsonnosliw-eb3yc]

loved this video, i've wanted to learn about logarithms since ~4 years ago, but i was apparently "not the right age" to understand everything. also that was a beautiful voice crack at 16:21

[@kaneed2769]

I've always loved Logs, they make complex problems comically easy. Also very interesting and a good tool in calculus.

[@sapuljeun7410]

Man you are crazy for writing all those digits. Such a great video.

[@richardnineteenfortyone7542]

Just for fun, teach yourself how to calculate square roots one digit at a time. the first couple of digits may be difficult, but after those, you will discover that the process can be greatly sped up with use of.... a slide rule! [No fair using a table of logarithms!]

[@WelchLabsVideo]

Lol was thinking the same thing when i did it wrong the first time and had to start over!

[@MdMatius]

Wake up, new Welch Labs video. We've been blessed today

[@imotvoksim]

The first minute is SUCH a nice way to show that taking a log of multiplication (division) yields addition (subtraction)! Love it!

[@WelchLabsVideo]

Woohoo!

[@philipwatson2407]

The good news is that slide rules are not entirely obsolete. They remain the quickest way to calculate the effects on performance and power consumption when changing the driven speed of a centrifugal pump. The flow rate varies proportional to the speed change; the head (pressure) generated varies proportional to the square of the speed change; and the power consumption varies proportional to the cube of the speed change. Thus, if you double the rotational speed of a centrifugal pump, its flow rate will double, its generated head will quadruple, and its power consumption will increase eight times. If you have the manufacturer's performance graph at any given speed, then a single setting of the actual speed against the graphed speed on the C and D scales will allow you to re-graph the entire performance and power consumption characteristics.

[@mattp422]

In college, I routinely used a slide rule and log tables. I was a senior when the first TI and (somewhat later) the HP calculator became affordable to a college student.

[@privacyvalued4134]

The mathematical properties of logarithms are extremely useful when working with incredibly large numbers that won't fit into a standard IEEE double on your average computer system _today_ and would otherwise overflow the limits of IEEE doubles when doing the calculations. You can do scaled math operations on the natural logs of values and then get the final answer by taking the exponent. Logarithms are _very_ useful in the field of statistics where you can be working with insanely huge numbers that would trigger NaN's all over the place. You can still get a NaN and be forced to use a large numerical library, which are more computationally expensive/magnitudes slower than hardware IEEE implementations, but that's a lot harder/rarer to come across.

[@samuelwaller4924]

could you give a simple example of this? I don't understand how the exponent (mantissa, right?) becomes the answer.

[@MattMcIrvin]

The 1999 video game Quake III Arena used a fast inverse square root algorithm (useful for normalizing vectors) that was famously mysterious. The key idea of it was that if you just cast an IEEE floating point number to an integer, because the exponent is in the most significant bits, that's like taking a really crap logarithm. You do your manipulations in that integer domain, cast back to a float, and you've got an answer that is not very accurate but you can refine it with some Newton's method iteration. The method is long obsolete because modern computers have more efficient ways of doing this kind of thing, but it was cute.

[@petercalkins3011]

Bravo! Fantastic job of bring logs to life. One on the best KZbins ever!!!!!

[@BenRasmussen-c3u]

this was very interesting. I used use a slide rule back in the 70's when I was in high school. I thought it was pretty simple. Then someone invented the HP calculator and I couldn't get my head around it, but everyone else loved it. Memories.

[@hanifarroisimukhlis5989]

Man, what a journey. I think i have a table like this in the "old books back when my parents were in uni" stack, along with National Semiconductor chips datasheets and anatomy textbook. Imagine if Briggs copyrighted his table, log tables would *never* be this successful. Kinda weird that maps somehow is copyrightable, but math tables aren't.

[@stevenlu7324]

Welch! So great to see you pop up in my feed! Loving this production value, wow!

[@WelchLabsVideo]

Thanks so much!

[@rayoflight62]

That curve happens to be the V/I curve of a forward-biased PN semiconductor junction. It allows the use of a simple diode to build logarithm amplifiers and converters. It is at the base of analogue computers...

[@WelchLabsVideo]

Nice!

[@glandeokrayo9956]

The base of analog computers is the operational amplifier, not the diode.

[@louco2]

One of the best videos on KZbin!

[@Athropod101]

Please upload more frequently-your videos are some of the best available on KZbin!

[@WelchLabsVideo]

Working on it!

[@donegal79]

That was amazing. KZbin really allows experts like Mr Welch to enlighten us all. I found this so exciting!

[@raulrueda1882]

Amazing and educactional. I will use this video with my math students this year.

[@WelchLabsVideo]

Awesome let me know how it goes!

[@raulrueda1882]

@@WelchLabsVideo I will!

[@hastypete2]

Thank you for the video. It is so important to know "why" things work to really appreciate how they work. I had a better comment, but it got mucked up by someone with a big head that had to tell me how wrong I was in a loooonnng post. I deleted my comment because I didn't want anyone else subjected to that. Whether or not I was wrong, that person overgeneralized my comment and went way too far to prove his superior intellect. It's humorous to me that I was able to delete his 30 minutes of ranting. Very refreshing.

[@Asterism_Desmos]

Amazing video, the content, story telling, videography, and (most importantly) the math! Phenomenal job, keep up the good work! p.s. We use the same calculator :D

[@flatfingertuning727]

I would think that one could use the fact that log((a+b)/2) = log(sqrt(a*b)) to produce a table of interpolated values until one reached the point where linear interpolation would be acceptably accurate. Alternatively, given a table of eleven values with the antilog of 1.0, 1.1, etc. up to 2.0, it's possible to compute logarithms at a cost of five multiplies per decimal digit of result. Given a value 1.0 to 10, the first digit of the logarithm will be 1 and the second digit can be found using the aforementioned table. Either divide by the largest table entry below one's value, or multiply by entry 10-k and divide by 10, to get an answer in the range between 1.0 and 10**(1.1). Then raise that number to the tenth power (compute its square, and the square of that, multiply those to get the fifth power, and square that to get the tenth power--four multipliex) to yield a value from 1.0 to 10, use the table to find the next digit, etc.

[@nomukun1138]

Nicola Tesla was said to do absolutely astonishing mathematics in his head, like calculating square roots and other stuff that seemed impossible, because he memorized logarithm tables!

[@chyldstudios]

Wow, you just opened my eyes to some new concept I didn’t completely understand.

[@benhur2806]

Wouldn't log tables also allow you to use Karatsuba's or a similar divide-and-conquer multiplication algorithm quite effectively? Assuming 5 digits of precision from a log table, even just 4 iterations of it, if a bit tedious, would give you 80 accurate digits...

[@justAI777]

Very well explained. Can you do a video on how you do your animations?

[@markniall3205]

This was a fantastic video, thank you for taking the time to make it.

[@WelchLabsVideo]

Glad you enjoyed it!

[@KipIngram]

I been suddenly struck by the pervasiveness of KZbin videos on logarithms - most presenting them as this utterly amazing and wonderful thing. And they are, of course. But before I started paying attention to this I just took it for granted that at least 80% of people walking around on the street would KNOW what logarithms were - I've always regarded them as part of basic high school math that everyone learned as they came along. It's a little sad that seems not to be the case.

[@timothykeech7394]

At school I used logarithms, at University and in my first job as an engineer, I used a slide rule and, nowadays when teaching I use a calculator, or, if that's too slow, mental arithmetic. It's nice to have all of them available. I don't have an abacus. Maybe I should look into getting one. I still have a set of tables (my dad's) and a slide rule.

[@cecilponsaing2749]

An absolutely delightful program.

[@SolathPrime]

I used to study industrial Technologies and this logarithmic notation is very useful not only on the digital scale but it goes to analog also

[@pocojoyo]

The best video on the history of logarithms

[@adarshkumar4336]

This was beautiful. Thank you.

[@dogcard664]

Would be great if you made another video linking how Logarithms help aid in the invention of the number e and/ or how this leads to the Natural Logarithm. Love Your vids!!! Keep it up💖💖

[@WelchLabsVideo]

Noted!

[@DJ-Eye]

Eulers number was THE only thing I felt was left out of this blast of brain-candy, but understandably may have waranted it's own video. It's a shame that 20 minutes seems to be the defacto standard for attention spans these days. I second the motion to "Keep it up".

[@musicalBurr]

Wonderful video! Thanks for kicking out another vid for us. We’ve missed you!

[@pedroth3]

Best explanation of this historical computation!

[@WelchLabsVideo]

Thank you!

[@MaximumBan]

WOW!!! Just WOW!!! I love math but knowing the history of its discovery is much more fun!!!😂 Thank you for this video! Epic!❤

[@CatFish107]

Well now you've got me jazzed to pick up a slide rule and practice with it. That was neat, thanks.

[@WelchLabsVideo]

Nice!

[@miguelf218]

This video was brilliant. Thanks, this is an amazing work.

[@varunahlawat9013]

Lovely video! I had always thought of these fundamental things that who the heck wrote logarithm values during the time of hand calculations, who found out the multiplication and long division methods that we use, ... Your videos are the best explainers of how did the modern mathematics come to exist.

[@WelchLabsVideo]

Thank you!

[@ke9tv]

A slide rule lived in my pocket from about 1970 to about 1975. (Before 1970, I was a bit young. After 1975, slide rules were dead and gone.) Because sometimes you just have to log a couple of rhythms! "What does the mantissa do in mathematics?" "Nothing. It just sits there like a bump on a log." It's questionable whether Briggs understood that you could use a second round of Napier's tables to divide your natural logarithm by ln(10)! He did far too much work because the formula log_a(b) = ln(b)/ln(a) was perhaps unknown to him. People were still wrapping their heads around logarithms. Nowadays, I find myself using ln(x) (and even log2(x)) much more often than log10(x). Common logs are mostly for reporting quantities in decibels.

[@Unique-Concepts]

I really love these videos. Thank you welch labs. The most interesting part in this video, showing the process of invention.

[@WelchLabsVideo]

Glad you enjoyed it!

[@letsimage]

7:54 amazing video thanks! But i think there on the table is better to write log (0.9) x

[@caladbolg8666]

Thanks for this video! I knew about the history, but I've never seen how the values in the tables were actually calculated.

[@ryvyr]

Thank you so very kindly for having non-adsense at either front/back/both rather than interrupting like TV, which feels regressive in the year 2024 ^^;; When mutually considered in placement, I then watch fully!

[@josleurs4345]

I thought that brigs had found log 2 by searching in the first table . He had taken 2tht 4th 8 root and so on , he had could have found it by division ... 2 devide by the number that is just under 2 ... then taking that number and deviding it again , ... that number finaly becomes 1 ... then he had to add the corresponding logs ... or by an itarating proces ... starting with log 1 en log 10 , taking squareroot , than you know that log 2 is between 0 and 0.5 , then geometric mean of 1 and sqrt(10) , and on the log site the arithmic mean ... , and so on ... thank you for showing the way he did ...

[@roberthuff3122]

🎯 Key Takeaways for quick navigation: 00:00 📐 *Introduction: Spotlight on a unique mathematical curve* - Introduction of the unique mathematical curve that transforms complex problems into simpler. - A glimpse how this curve maps mathematical operations from one axis to another. 01:17 ⚓ *Applying the curve in nautical navigation* - A practical example of navigation while crossing the English Channel. - Calculation of course using sign tables available in the 1600s. 02:24 📖 *John Napier's revolutionary book* - The publication of John Napier's book, introducing Logarithms. - The book contains tables of signs and a column labeled logarithm, to simplify navigation calculations. 04:17 🔢 *Broad application of Napier's logarithms * - Explanation of how Napier's logarithms impacted and simplified computation. - The East India Company started leveraging Napier's method due to its speed and accuracy. 05:24 🧮 *The birth of computational tools* - The creation of computational methods and tools such as tables of logarithms and the slide rule. - These methods ruled the world of computation for nearly 350 years until the advent of electronic calculators. 06:21 🔠 *Understanding the mechanics of logarithms* - A deep dive into understanding how Napier's numbers or logarithms work. - Construction of a miniature version of Napier's table for demonstration purposes. 09:21 🧩 *Filling the gaps to expand logarithms' usefulness* - Addressing the limitations of our simple values and miniature Napier's table. - How Napier constructed his table using a base of 9999999, making it dense and useful for a wide range of numbers. 10:43 🔄 *Revolutionizing logarithms: The tweaks by Henry Briggs* - The story of Henry Briggs visiting Napier and discussing the improvements needed in Napier's tables. - Transformations Briggs proposed for creating logarithms in a simpler and more useful way. 13:39 🔬 *Zooming in on the curve for better computations* - The ingenious method Briggs found to compute logarithms by zooming in on the curve. - Understanding the principles of logarithm computation by taking repeated square root operations. 16:39 🛠️ *Crafting base 10 logarithm tables by Briggs' approach* - How Briggs computed logarithms of numbers using powers of 10 and his new linear formula. - Overcoming complexities by exploiting characteristics of the decimal system. 18:00 🌐 *Logarithms' essential role in computation and calculation* - The significance and wide application of logarithms as an indispensable tool for computation. - Briggs and Vlacq's logarithm tables largely unchanged and used till the 20th century. 18:15 🧪 *Slide Rules: The practical application of logarithms* - Explanation of how slide rules, derived from Briggs and Vlacq's logarithm tables, were used to perform fast yet precise calculations. - Demonstration of multiplication, division, and square root operations performed using a slide rule. 19:20 📊 *Mapping powers of two on the slide rule* - Examination of how powers of two are represented on the slide rule. - Explanation of how the spacing on the slide rule matches the curve where multiplication by two on the x-axis equates to addition of one on the y-axis. 20:16 📚 *Slide Rules and Logarithms: A powerful alliance* - Discussion of how the slide rule uses logarithmic spacing of tick marks for calculation. - Breakdown of how multiplication and division are performed on the slide rule. 21:13 💻 *How the Electronic Calculator replaced the Slide Rule* - Introduction of the HP35 electronic calculator in 1972, which made slide rules obsolete. - Description of how HP engineers utilized Brigg's calculations in the development of the electronic calculator. 22:08 🌎 *Logarithms: The most potent Computing instruments in History* - Highlighting the significant contribution of logarithms in science, math, and engineering for 358 years. - Explanation of how logarithms quickly disappeared from daily use after the invention of the electronic calculator. 22:41 💡 *The enduring Importance of Logarithms* - Despite the decline in the use of logarithm tables and slide rules, logarithms' importance in fields like machine learning and data visualization is mentioned. - Recommendation of Brilliant's courses for learning about logarithms and their applications. Made with HARPA AI

[@donsanderson]

Really excellent video. I loved the pacing

[@glandeokrayo9956]

I still have my log tables and slide rule from my time in school in the late 1970's and early 1980's . Interestingly, the problem now is not the manual calculation of multiplication or division, but addition and subtraction! If you had an 'addiator' you were set. The addiator was invented in the mid 1800's and allowed for quick addition ans subtraction calculations.

[@petrospaulos7736]

you are back!!!!

[@kenkiarie]

Always a pleasure! Thank you.

[@learningpiano]

thank you for such high quality and highly dense knowledge content

[@josefsedlak7605]

I was looking for this video since I have seen bits of it on Tik Tok. Going to show it to all my math students and colleagues as well. Splendid job Welch Labs.

[@WelchLabsVideo]

Awesome thanks for watching!