Why Logarithms Appear in This Integral

Before the days of Calculus, one Pierre de Fermat wanted to find the area under the function f(x)=x^n. This problem we now call "integration" was then called "quadrature" or "squaring". Fermat was able to square every function f(x)=x^n for any rational n except for one case: n=-1 (that is, the hyperbola). It turns out that this unique nature of the hyperbola was tied to logarithms and Euler's number e. But why? Why does the area under the hyperbola have anything to do with logarithms or e?
Other resources
e: The Story of a Number by Eli Maor
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Music
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Starboard by Patricia Taxxon

[@jHan]

Mistake at 7:17: log(1)=0, log(10)=1, log(100)=2, log(1000)=3

[@comic4relief]

Glaring mistake.

[@mohammedbelgoumri]

In his video on exponentials and logarithms, Grant from @3b1b asks if anyone knows of a beautiful geometric interpretation of logs. Here it is.

[@Zosso-1618]

This is amazing! I remember reading various proofs that the function describing the area under a hyperbola has logarithmic properties. But in one sentence, you made that connection deeper than my reading did! Bravo!

[@guillaume5313]

Great content as always !

[@TheMemesofDestruction]

Fantastic decomposition!

[@DavyCDiamondback]

Really, it's ln(|x|)+C, coincidentally, no matter what C you choose, if this function intercepts y=x^2, it intercepts perpendicularly

[@matiasvergara5069]

Very nice video jHan!

[@Avighna]

The video is amazing, but the title isn't as great as it could be. Something like "How Descartes discovered the power rule without calculus" might be better, anyway amazing job man!

[@davidwright8432]

True, but that drags Descartes into it, and in terms of this explanation, he's irrelevant. Interesting guy, tho!

[@Zosso-1618]

Did you watch the video? Descartes isn’t mentioned once, that was Fermat. And most of the video is about the area under the hyperbola, not the power rule.

[@eduardoguimaraes5226]

Do you use manim?