What makes the natural log “natural”? | Lockdown math ep. 7

From 3Blue1Brown.

All about ln(x).
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Beautiful pictorial summary by @ThuyNganVu:

At minute 16, the sum should be written with a "…" to indicate going to infinity.
At minute 38, the exponent should have 1/(2s^2) instead of 1/s^2 for s to represent standard deviation.
At minute 54, an equal sign was mistakenly used in taking the derivative of x^3 / 3!.
At the end, it should be pointed out that the alternating series with x^n terms only converges for values of x between -1 and 1, so the values one can’t be considered proven with values of x outside that range. Everything with the argument here is fine, as it only deals with the convergent input, but that fact should still be mentioned.

Related videos.
Calculus series:

The sum giving pi^2 / 6:

The sum giving pi / 4:
https://youtu.be/00w8gu2aL-w (Mathologer)

Video timeline (thanks to user "noonesperfect")
0:00:14 – Question 1
0:02:29 – Answer 1
0:06:27 – Prime nos. in Infinite Geometric Series (Basel problem) and their relationship with Natural logarithm
0:12:01 – More examples of prime numbers in infinite series and their relationship with ln
0:17:25 – Question 2
0:19:20 – Answer 2 and explanation using ln
0:22:25 – Question 3 and families of curves
0:26:37 – Answer 3 and explanation
0:28:50 – Imaginary exponential
0:30:57 – Derivatives of exponential terms
0:37:21 – Why derivative of e^t is the same as that e^t itself?
0:41:21 – Question 4
0:44:12 – Answer 4 and explanation using Python
0:46:02 – Taylor Series for e^x
0:48:29 – Derivatives of polynomial terms/Derivatives of e^x
0:50:56 – Derivative of natural logarithm using graph
0:56:07 – Question 5
0:57:37 – Answer 5 and explanation
1:02:15 – Euler–Mascheroni constant
1:08:37 – Question 6
1:12:41 – Connecting dots to the familiarity of different expression in math

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Music by Vincent Rubinetti.
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