- 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"

-12:36 I realised after publishing the video that the method I used was not the method that was used by Saint Vincent. For the original method by Saint Vincent check: https://www.jstor.org/stable/3620207

-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.

- 7:39 Napier published two books regarding logarithms: the Descriptio which describes how to use logarithms and the Constructio, which describes how to construct them (published posthumously). Modern English translations for both books are available at: http://www.17centurymaths.com/

- 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.

- 9:16 The link to how Oughtred and Speidell came up with logarithms that resemble natural logarithms ​​https://www.tandfonline.com/doi/full/...

- 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.

-17:29 The statement: “It is not clear why Mercator called it the natural logarithm” may raise eyebrows as the following three ideas are usually given as a reason: 1. e is a quantity which arises frequently and unavoidably in nature, 2. Natural logarithms have the simplest derivatives of all the systems of logarithms and 3. In the calculation of logarithms to any base, logarithms to the base e are first calculated, then multiplied by a constant which depends on the system being calculated. See https://www.jstor.org/stable/3028204. However none of these ideas were known at Mercator’s time and there must have been a different reason as to why he called it the natural logarithm.