The History of the Natural Logarithm - How was it discovered?

- Learning about the history of the natural logarithm helps us understand what it is.

Today we define the natural logarithm as a logarithm with the base e and many people, understandably, wonder why! Interestingly the natural logarithm was discovered decades before the number e. In fact it was discovered before the link between logarithms and exponentials was recognized!

In this video I talk about how and why logarithms were invented, how the natural logarithm arise from from logarithmic tables without the need of the number e and how studying the area of the hyperbola was instrumental in defining logarithms and the natural logarithm.

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Corrections:

- 11:33 The name of the hyperbola is: "Rectangular Hyperbola" not "Square Hyperbola"


-15:13 Saint Vincent didn't make the connection with the logarithms, he only discovered that they areas are equal. It was de Sarasa who made the connection

Notes:

- 00:02 I am using the terms discovered and invented in a specific sense. “Discovered” when a mathematical idea was stumbled upon, while “invented” when it was created on purpose to solve a particular problem. Hence logarithms were invented, but natural logarithms were discovered.

- 00:03 It is difficult to pinpoint the exact date the number e was discovered. It is suggested that it was discovered by Jacob Bernoulli in 1683 while studying a question about compound interest. However Bernoulli didn’t find the value of the number nor did he link it with the natural logarithm. It was in 1748 that Leonhard Euler found the value of e and understood that it is the base of natural logarithms. And since the natural logarithm was discovered in 1647 through the work of Gregoire de Saint Vincent and Alfonso de Sarasa, it was therefore discovered between 36 years and 101 years before the discovery of the number e.

- 2:28 Some sources credit Jost Burgi with the invention of logarithms. Although there is evidence that he did come up with a similar idea before Napier, he didn’t publish it till after the success of Napier’s logarithms. Also, Burgi's approach was different and doesn't lend itself to understanding the natural logarithm.

- 07:09 I omitted the decimal points, the correct values are 9999998.0000001, 9999997.0000003 and so on. While Napier included those decimal points when he calculated the tables, he omitted them in the final result.

- 7:09 For Napier’s logarithms to work with multiplication and division, we have to divide by 10^7.

- 8:06 There is a lot of conflicting information about how Oughtred calculated his logarithms, “Wikipedia number e” page suggests that he calculated them from the number e, which is highly unlikely since the number e, as mentioned in the notes above, was not discovered till at least another 65 years, and even then, it’s value wasn’t calculated till yet another 65 years. A source I found most convincing is in the note below.

- 9:20 The way I explain it is not how Oughtred and Speidell calculated their tables, however, it gives a better understanding of what the natural logarithm is.

- 9:33 The smaller the base, the closer the result is to the natural logarithm. I chose 1.000001 to match the results of Oughtred and Speidell.

- 9:39 Again, I omitted extra decimal points, the correct values are: 1.000002000001, 1.000003000003 and so on.

[peterclancy]

Finally, after 72 years, someone explained “ where and how” about logs. Thank you so much!

[samisiddiqi]

12:36
This is for my own reference.

Hands down the best and most complete video on logarithms I've ever seen on YouTube. History and use included. I like it.

[jaytravis]

I've been an enthusiastic math hobbyist for years now and I've watched hundreds if not thousands of hours of math lectures. This is one of the most concise and original math explorations I have ever seen.

Thanks a million

[ishfaqmath]

I have been teaching mathematics in Asia for about two decades and I highly recommend students to watch extraordinary efforts by you to give a logical insight to basic concepts!
Highly appreciated and keep on unlocking the basic mathematical concepts as they are not really taught at high school and majority of students are unaware of them!

[nickxyzt]

I've been waiting for this explanation my whole life. I always wondered how on earth natural logarithms were invented before the discovery of calculus and the number e. This is the most outstanding video about logarithms ever. Well done!

[ingGS]

Beautifully explained. I am glad YouTube recommended you, great video!

[jonasgunnarsson]

The slide-rule, which is a logarithmic ruler with sliding parts, is an alternative version of the logarithmic table that utilizes this effect of logarithm to transform between multiplication and addition or between division and subtraction. As mentioned in the video, people used logarithmic tables and slide-rules to do calculations, before electronic calculators were invented. My mind was blown away when my dad showed me a slide-rule and I thought we should be taught about this. Thank you for the video, showing how this was invented.

[pashi]

I think the story-telling approach of explaining something complex like the natural log is very powerful and it should be done more regularly in schools. Well-done!

[bhanuinani]

The best explanation I could ever come across. Brilliant way to teach. The graphics were simply superb.

[samueldeandrade]

This man probably is the absolute BEST math youtuber we never had.

[qelmnl]

You really deserve a bigger audience. What is fascinating in mathematics is the line of reasoning that mathematicians follow to arrive at a concept. More so when the reasoning is rooted in experience and application. This is true intellectual discovery, and unfortunately far too often omitted form math class. Thank you for this video, I enjoyed it a lot and immediately subscribed your channel. Looking forward to more videos from you!

[chibbi]

Incredible!! Truely fasinating. Please create more videos on maths and science. As someone who requires the entire story on a concept to fully understand it, your content is exactly what I need!! You are an amazing teacher, thank you Tarek. Looking forward to more videos!!

[iyannazarian]

Words won’t do justice to how many time I got “wowed” by these great and powerful explanations! That’s 3b1b quality right there! You’re still at a 1000 subs, but remember me when you get that 100K ✌️

[joeaverage]

Best explanation on logarithm I've seen on youtube. Please make more of these kinds of videos please!!

[fredspiessens]

Thanks! I think the exceptional quality of your presentation stems from the way you combine clear examples and explanations with the actual history of how/when things were discovered. That makes it so easy to understand that teachers all over the world should probably copy your approach.

[matthewphares]

Bravo! One of the most enjoyable videos I’ve watched on YouTube. I think math would be easier to understand if first taught from the historical perspective of man progressing to president day.

[techdoc]

I really loved your video. I never knew the origins of logarithms and you presented it like a professional documentarian with a clear flow and examples. Excellent work!

[nkbrahm]

Never had these insights about logarithm..by far the best video I have seen on you tube..thank you so much

[geonalugala]

Fantastic! It is amazing that I can now create logarithms of my own at home, accurately, instead of googling everything. Thank you for this history. Few, very few people could have explained what you just did. Genius mind, sir.

[horseloverfat]

Totally awesome explanation. As an amateur, I have been teaching myself calculus via history & philosophy of science (my specialist field) and this video filled in a few more of the historical blank spots between Archimedes and Euler. This was clear, logical and concise - thanks so much!