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)

This is episode 6 in a series called Tricky Parts of Calculus, a series about the subtle and difficult parts of calculus that are usually glossed-over in a calculus class. Tricky Parts of Calculus playlist: https://www.youtube.com/playlist?list=PLw3pvR_YJeRcMaubDZvkjayqDJT4Tx47A

Also check out my channel for general advice and opinions about math, as well as the Daniel Rubin Show podcast for interesting conversations about math and other topics: https://www.youtube.com/channel/UCYLOc-m8WuQaBHJ56lEw4VQ

For a discussion of how e^x shows up in analysis of the Martingale betting strategy, see this great video from Numberphile: https://www.youtube.com/watch?v=zTsRGQj6VT4


References:
Hald, A History of Probability and Statistics and Their Applications before 1750 https://amzn.to/2ThUmip
Havil, John Napier: Life, Logarithms, and Legacy https://amzn.to/2UpMjR7
Newton, Principia https://amzn.to/3ikO5Ln
Toeplitz, The Calculus: A Genetic Approach https://amzn.to/3kvNwRE
Williams, From Sails to Satellites: The Origin and Development of Navigational Science https://amzn.to/3hLHY3w
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