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

From 3Blue1Brown.

All about ln(x).
Full playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev
Home page: https://www.3blue1brown.com
Brought to you by you: https://3b1b.co/ldm-thanks

Beautiful pictorial summary by @ThuyNganVu:
https://twitter.com/ThuyNganVu/status/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:
https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr

The sum giving pi^2 / 6:
https://youtu.be/d-o3eB9sfls

The sum giving pi / 4:
https://youtu.be/NaL_Cb42WyY
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

——————
The live question setup with stats on-screen is powered by Itempool.
https://itempool.com/

Curious about other animations?
https://www.3blue1brown.com/faq#manim

Music by Vincent Rubinetti.
Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown

Stream the music on Spotify:
https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u

If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.

——————

3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe

Various social media stuffs:
Website: https://www.3blue1brown.com
Twitter: https://twitter.com/3blue1brown
Reddit: https://www.reddit.com/r/3blue1brown
Instagram: https://www.instagram.com/3blue1brown_animations/
Patreon: https://patreon.com/3blue1brown
Facebook: https://www.facebook.com/3blue1brown