What makes the natural log "natural"? | Ep. 7 Lockdown live math

All about ln(x).
Full playlist:    • Lockdown math  
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Beautiful pictorial summary by @ThuyNganVu:
  / 1259288683489849344  

Errors:
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:
   • Essence of calculus  

The sum giving pi^2 / 6:
   • Why is pi here?  And why is it squared?  A...  

The sum giving pi / 4:
   • Pi hiding in prime regularities  
   • Fermat's Christmas theorem: Visualising th...   (Mathologer)

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld

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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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