How people came up with the natural logarithm and the exponential function #SoME1

I discuss the history of the introduction of the natural logarithm and exponential functions, answering the question of how the logarithm was first discovered and how it's properties were derived, and also why it appeared when it did. I consider several other areas of inquiry that could have led to a theory of exponentials and logarithms, including: compound interest, population/economic/technological growth, physics, gambling, navigation, and pure mathematics. Then I explain the work of John Napier in constructing calculation tables for astronomers that introduced the natural log in 1614.

0:00 Intro
1:05 History of compound interest
6:55 Why exponential growth was not a recognized feature of life in the past
8:10 Exponential functions in physics
10:58 Exp and log in gambling
14:34 Navigation, meridional parts, and the integral of the secant
18:43 Calculation tables and Napier's introduction of ln(x)

References:
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[DanielRubin]

I forgot to mention the Fibonacci sequence! Leonardo of Pisa (Fibonacci) described the sequence F(0)=0, F(1)=1, F(n+1) = F(n) + F(n-1) in his book _Liber Abaci_ from 1202 in the context of a word problem about the breeding of rabbits. The sequence was known in Indian mathematics centuries earlier. The sequence does grow exponentially, but the formula F(n) = 1/Sqrt(5) (((1+Sqrt(5))/2)^n - ((1-Sqrt(5))/2)^n) was only written down in 1843 by Binet.

[mgostIH]

I loved this! Some more history of math should be taught in high school when introducing concepts, we take for granted a lot of impactful work that really had centuries of refinement and adjustement that now seems given by definition on a blackboard that has no relation to reality.

[jimmyzhao]

People have always been so smart. These origin stories are fascinating.

[jaybrown]

Since I last studied math for engineering about 45 years ago I had a very hard time keeping up with this video but I still find it fascinating that people were able and willing to produce such complicated work hundreds of years ago. I often wondered how and where all the log tables came from. I think I was one of the last engineering classes that used the slide rule for some of our classes.

[ratguy]

I don't get to learn about this part of math, the historical part, in school but it gives me so much more vigour and interest in mathematics. Thank you

[macrumpton]

When I asked my math teacher where did these magical logarithm numbers come from, she said they are from the back of the math book.

[khj]

I believe math should be learned based on problems that need to be solved, rather than blindly memorizing formulas. So, to solve a specific problem, you'd try out many different methods with increasing sophistication, by first using the method used in the stone age, then with the method used in the year 1600, etc, until you arrive at the most efficient/advanced method, which also provides the correct answer. This way you acquire real knowledge and understanding of what you're doing.

[paulklee]

Greetings from the U.K....Thank you for this... as a layperson interested in maths, all to often actually uses and examples are not talked about and even rarer the historical background to mathematical concepts... this helps immensely! Though I must say that even now, fifty years after leaving school I’m still waiting for that moment when a knowledge of trigonometry will save my life, as I was constantly told it would!

[asifmahmud]

Came across this gem today.
I cannot express enough, how fascinating these contents are .
Thanks a lot

[federicocarrone]

Daniel this is amazing. I will have to watch it many times to process everything. Thanks for taking the time to create this.

[kdicus]

Absolutely LOVE the history weaved throughout. Amazing, amazing video. Please do more - you have a gift for mathematical story telling. There are more and more people coming who speak the language of math - and this quality of video raises the bar. Reach out.

[zhess]

History, math, economics, and physics in one video. I love it

[johnnytoobad]

When Math was taught to me...Nobody paid much attention to...or even talked about the history. Great Stuff.

[techconbd]

History of such kind is so rare...please keep uploading more videos like these...
Love from Bangladesh 🖤🖤🖤🇧🇩

[katalyststem]

@ Daniel Rubin: what a fantastic 30 mins. I can appreciate the amt. of research and hard work that has gone into this.
I have History of Science and math integrated into my STEM course for school students and 2 days are devoted to this entire subject (devoted to relationship between e and compound interest; the need for logarithms etc. ). this video has given me a refreshingly new perspective to make the class more enjoyable.
Thanks a ton for this video

[vvop]

Hello from Adelaide, South Australia. I came here for the maths, but the history is just as interesting, thanks.

[GoSlash]

I actually have a slide rule collection. 😅 I find them superior to electronic calculators in many ways, especially when the slide rule is specialized to perform a specific job. There are slide rules out there specifically designed to compute the frequency of resonant circuits, perform dead reckoning navigation, convert between Imperial and Metric...

[travboat]

I've wondered about this for so long. I've always guessed that e^x was intentionally created as a means to have a function that equals its own derivative

[yakuzzi]

This is such an important video. I mean the historical context in which the maths was developed really helps one understand in which other contexts they are applicable.

[postxian]

Another application of the logarithm not mentioned here is the well-tempered musical scale. Interesting coincidence that the invention of this scale was contemporary with invention of the logarithm.