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

All about ln(x).

Beautiful pictorial summary by @ThuyNganVu:

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:

The sum giving pi^2 / 6:

The sum giving pi / 4:

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

-------------------
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.

Curious about other animations?

Music by Vincent Rubinetti.
Download the music on Bandcamp:

Stream the music on Spotify:

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.

------------------

Various social media stuffs:

[NUSORCA]

“To see if you have been paying attention so far” this is the scariest thing to hear in math class

[TheBookDoctor]

That riff on 69 was just about the smoothest damn thing I've ever seen.

[SoumilSahu]

This dude is an absolute legend for the following reasons:
1. Amazing, well thought out, quality lessons.
2. Trolling the 69 trolls.
3. DAT GORILLA

[tetraedri_]

27:40 I love how he just "happened to know" the value of ln(69). Not that he anticipated this or anything.

[sibiakkash]

The planning of the '69' baffles the war strategists to this date.

[evanjohnson]

26:40 only Grant could make a video targeted to high schoolers, get 69 as the answer to their favorite number, and with a straight face innocently deliver a flawless math fact as to why 69 is interesting which has absolutely nothing to do with why 69 is up there lol love this channel 😂

[bluebrown]

Thanks to everyone who joined! One thing I should add: For the manipulations at the end, all of that is only valid for values of x where the relevant series converges. So as x gets outside the range [-1, 1], it wouldn't necessarily follow that you get similar results.


Edit: Once again, bizarrely, after I cut out the intro animation so that the video actually starts where the lesson begins, despite a preview in YouTube's editor indicating otherwise, the end got chopped off. It's only a half-sentence with little significance, but if you're wondering what's going on, well, there you go.

[RobertLeyland]

The gorilla was awesome!


Plus the way Grant was prepared for 69 was great. Completely trolled the trolls in the audience.

[noonesperfect]

0:14 Question 1
0:57 Python Program to find all primes in a range
2:29 Answer 1
3:02 Explanation (Mathematician logic to find prime nos. in particular range)
06:27 Prime nos. in Infinite Geometric Series (Basel problem) and their relationship with Natural logarithm
12:01 More examples of prime numbers in infinite series and their relationship with ln
17:25 Question 2
17:58 Can you spot Gorilla?
19:20 Answer 2 and explanation using ln
22:25 Question 3
22:46 Does family of curves depend on any particular mathematical constant?
26:37 Answer 3 and explanation
28:50 Imaginary exponential
30:57 Derivatives of exponential terms
37:21 Why derivative of e^t is same as that e^t itself?
38:10 Explanation using graph and limit
41:21 Question 4
42:32 Audience questions from tweeter
44:12 Answer 4 and explanation using Python
46:02 Taylor Series for e^x
48:29 Derivatives of polynomial terms/Derivatives of e^x
50:56 Derivative of natural logarithm using graph
56:07 Question 5
57:37 Answer 5 and explanation
1:02:15 Euler–Mascheroni constant
1:03:30 Alternate expression for infinite series
1:08:37 Question 6
1:09:41 Answer and Explanation
1:12:41 Connecting dots to familiarity of different expression in math / Genius way to approach solution

_Water sips_ : 18:48, 18:54, 42:04, 56:36, 1:09:10

*Note: Changes were made according to new time-line, Let me know if any errors or changes you think need to be made. Thanks*

[CrypticPulsar]

I'm 50 years old.. I never got a chance to learn logs in high school in Iran in 1987 because Saddam was lobbing missiles at Tehran and schools closed.. logs always remained an unlearned topic for me and much later here in the US and throughout college I never truly grasped the concept, particularly e and natural logs.. I've watched your videos many times and I keep watching them and I love them because finally and after all these years I'm getting it, thanks to your beautiful explanation and these awesome demo tools. Thanks a million!

[Vaaaaadim]

26:45 The man prepared for this eventuality by thinking of something beforehand that would yield the number. Well played.

[thadsuits]

If you like the joke about the logorithm neperien, maybe you'll like this one:
Q. What does the B in "Benoit B. Mandelbrot" stand for?
A. Benoit B. Mandelbrot

[Autmatin]

An infinite number of mathematicians walk into a bar. The bartenders ask what they want to drink. The first one walks up to the bartender and says, "Give me a beer." The second chimes in, "I'll take the same, but half." The third says "Same, but give me a quarter." As the fourth mathematician starts to order the Bartender holds up his hand, turns around, pours two full beers, walks back to the counter and says, "You guys really need to learn your limits."

[MasterChakra]

42:55 The terrible pun comes from "népérien" sounding almost like "nepérien", which is homophonic to "ne paie rien" meaning "doesn't pay anything".
So e has to pay the beer because logarithm (in base e) "doesn't pay anything".

[AndreaBraghiroli]

You really make me feel beauty in numbers. Never posted a comment on YouTube before, but I really need to thank you for all this.

[PetWolverine]

I just noticed that Grant SOMETIMES crosses his 7's.

Look, I can get on board with crossing them or not crossing them, but crossing only SOME of them? What madness is this?

[djd]

I remember when we learned about prime numbers in primary school, we had a homework assignment to figure out all the prime numbers from 1 to 100. I wrote a program in BASIC to do the assignment for me. In my mind, it told me that I not only understood the concept, but also understood it enough to explain to a machine how to understand it. I made the mistake of telling this to my dumbass teacher and she failed my assignment, on grounds of "cheating". This was back in the early/mid-90s. I wonder if teachers these days would still have the same attitude.

[nickpro]

At home I have electronic scales which show your weight in kilograms, and I often see the number 69.3 kg and I am always like "yay, the natural log of 2 once again!"

[kentuckysugarbear]

As a mathematics major who hasn’t used his degree in years, thank you! I love these! Reminds me of how much I used to love math in school.

[gcewing]

I don't buy this French stuff. Clearly, "ln" stands for "latural nog".