What is e and ln(x)? (Euler's Number and The Natural Logarithm)

Euler's Number, e, is one of the most prominent constants in mathematics and exponential functions are some of the most important in maths. In this video: we take a deep dive into what Euler's Number is, what exponential functions are and how to differentiate them and we take a look at the super important natural logarithm, ln(x).
Website: mathacy.com
Timestamps:
0:00 - Intro
0:27 - Compound interest
2:37 - Defining e (Euler's Number)
3:30 - Differentiating exponential functions
6:10 - Derivative of e^x
6:48 - The Natural Logarithm - ln(x)
8:22 - Derivative of ln(x)

[@angrwl]

Spent hours trying to understand what is eulers constant and got it straightaway by looking at this. Thanks a lot.

[@samueljehanno]

Good

[@vortex6132]

That’s great but in terms of its properties I’d go much further

[@Premium55]

​@@vortex6132any good videos?

[@abbasfadhil1715]

I rly understood every thing in this vid but the factorisation of p

[@nimo517]

I’m looking for one of those in regard to quantum theory and the applications in real world. I understand so far _it’s used on a micro scale when viewing the world, and small objects like atoms, and how the fundamental laws in nature do not apply any longer at the quantum level._ and I understand that concept, but that’s where I’m left in terms of my true understanding and faith in quantum theory. After describing the basic desires and principles of the study, I don’t really understand how the study is supposed to translate to real life. Then the thought experiments like the *cat in a box*? I sometimes think it’s just mental master***tion and all hypothetical talk. Sure I believe it’s provable on paper but to me I’m sooo close to getting it, but until then it’s like being a kid trying to pray and not knowing what to do

[@sw-reload9232]

The explanation feels like getting an eureka moment every 5 seconds or so. Realizing all the connection you've learned in math

[@manavgupta2059]

Fr!

[@finleycooper8844]

Just a heads up 'e' is Euler's number. Euler's Constant is a different constant (Also called the Euler-Mascheroni constant sometimes) and is denoted by a lowercase gamma

[@JunioSonicT.Hedgehog]

and it is a part of the connection inbetween the Harmonic numbers and Factorials, so that's neat.

[@69erthx1138]

Euler number, Napier constant is e. Oily-Macroni constant is gamma.

[@ChatGPTGamingReal]

@@69erthx1138oily macaroni

[@newsgo1876]

It is worth emphasizing that "e" is a number defined through the limit operation. And it happens to show an unique property in another limit operation, i.e. the derivative, that e^x has itself as its derivative. This is one of the wonders associated with the limit operation and the infinity process. Essentially, a number is just a symbolic name for a bunch of operations, except the symbols for the natural number 0~9.

[@leslyvalerianavarroberrone9323]

But what does it mean its derivative it's the same as the original function? As I knows, the derivative shows the growth rhythm for that function. Does it mean that the growth of f(x)=e^x is constant?

[@dipteshrai5979]

But 2 is an operation of 1+1 and 0 is 1-1. Is it not? And 1/ 2 is 1 divided by 1 + 1

[@newsgo1876]

​@@dipteshrai5979 Profound! So it seems there's nothing fundamental beyond the number 0 and 1. Does this imply the binary representation?

[@dipteshrai5979]

@@newsgo1876 well, I said that 0 is 1-1. So I’m saying, only 1 exists. Unless u want to think of 0 as an absence of 1, or an absence of anything.

[@newsgo1876]

@@dipteshrai5979 Yes, your idea is very impressive! I agree with you.

[@user-dy9nv4os4q]

I WAS VERY HELPLESS ON KZbin TO FIND OUT A RIGHT VIDEO TO LEARN THESE LESSONS. THEN THIS ONE CAME. THANKS A LOT. LOVE FROM INDIA

[@LEKSANDER01]

I am in Pre-Calculus and have failed the class twice already, but this third time I am dedicated to passing... not only passing but understanding everything well, like you said, inside and out. I struggled with logs, exponentials, ln/e, but it's no longer petrifying. Such good animations too!

[@staticion598]

💀

[@ethinesvedi7246]

Well then buddy , best regards, understand the concept dont rush to obtain marks ,best of luck

[@chitrakshsingh5661]

then just you wait till you get to actual calculus :skull:

[@LEKSANDER01]

@@chitrakshsingh5661 I'll do well, I failed twice because I was severely lacking in basic algebra... it is pretty funny lol

[@make-a-wish2224]

you hell💀

[@karimjanmohamed3835]

Doesn’t get better than this when it comes to explanation. Thanks!

[@kennethchao1290]

anyone still confused here?

[@user-ef9zk6np2e]

Yeah me😅

[@manavgupta2059]

Where did u got confused? I legit took 3 hours understanding this whole concept from this video and ran into lot of misconceptions So, I recommend u the following before watching the video: 1) Be expert with all logarithmic properties. (Solve 10 questions of each to master them) 2) Be expert with all exponential properties 3) Most importantly, learn differentiation first Then u will get second half of the video It is very complicated but once u understand it, it becomes very very simple :)

[@user-ef9zk6np2e]

@@manavgupta2059 no need now! I have understood it well

[@saravananrajendran9813]

Me too

[@notreallydavid]

Always!

[@ilove.science]

Finally a clean video Thanks!

[@emmanuelonah4596]

Wow! You virtually connected various aspects of mathematics in a very interesting way! Thank you!

[@aleenavirdee5656]

I love your videos! Excellent explanations - they never made this clear in school! Thank you so much!

[@indisingh2990]

Thank you Mathacy !! You explained this really well 👍🏼

[@jamesbentonticer4706]

This is great. I never understood where the decay function y=e^-×ln (2) actually came from! Fantastic thanks.

[@0x90meansnop8]

Mathacy, thank you so much. I ve been diggin from explanation to explanation without getting it.

[@aaronnanoo1887]

Brilliant video, cleared all my doubts 👍

[@jesca4855]

this was so insightful, love all the connections you made, really helped me put it all together.

[@joelwyatt9949]

Amazing video, you managed to perfectly explain e and ln in 12 minutes.

[@Jee2024IIT]

So much potential in this channel as the way he teaches

[@DiLLZGFX]

Great video ! Learnt a lot and beautifully explained !

[@shyyongc.9477]

Very well explained video! Made the "e" great again, and really this should be taught in colleges/unis to get students understand the intuition behind

[@leslyvalerianavarroberrone9323]

Yes, most students never know where does this stuff comes from, as " but that way they can't see the beauty in it.

[@henrynwosu6277]

Omo. I hope I get to meet this man someday and thank him personally. Because you're out there giving meaning to my engineering degree and helping me see these things intuitively. Allowing me to chase my passion for Mathematics. Thank you isn't enough. Hopefully I can do something for you too. Thanks @Mathacy

[@krishshah2209]

Another Masterclass as always! Thank You!

[@rickintexas1584]

I wish I had KZbin when I was in college. I graduated with a BS Aeronautical Engineering in 1986.

[@amirjutt0]

Boy you make incredibly awesome videos😻 Awsome explanation.I suggest you to be active on this channel and gradually you'll be listed in top class KZbinrs.

[@JASPREETKAUR-eo9oe]

I’ve had problem with this concept.But after watching this, all my doubts are cleared now. Thanks a ton.

[@forpublicstuff728]

Brilliant explanation! Thank you. It clicked when you said that we want 1 as a constant.

[@davindersehmi33]

This video was great! Very informative!

[@sathyanarayananviswanathan3770]

I bet this is the best video about mathematics on youtube. I didnt even skip a single second, I enjoyed it entirely.... You just earned a new subscriber.

[@manavgupta2059]

It is not the best in any way lol 3blue1brown is best Now what are u giving me because u lost the bet?😊

[@Dinesh123lol]

Amazing video, it helped me so much!

[@nagarjun385]

Damn!!! These many years as a STEM Student, I've never ever got the finest and the most intuitive explaination of Euler's Number. Great video Mathacy. Good Luck!!

[@exoticcoder5365]

this video is very useful, very very good, I hope it gains more recognitions

[@Uncreeperble]

Bruh if you had to calculate p + p is 2p using factorization there is a problem 😅😅

[@ronm3245]

You might have missed: That first calculation is intended to help establish the pattern. After all, math is really just pattern recognition at its core.

[@block7059]

Writing a test in a week about this topic, very nicely summarized

[@jayan99]

Great work, things unravel like magic, amazing and precise. Thanks again. Differentiation of e^x, lnx, a^x . Nice and smooth

[@ranjithamarakoon8842]

Excellent explanations. Thank you very much.

[@amrishravidas9258]

This video was super informative and it goes along great with 3Blue1Brown's video on Euler number. Also, at 6:26 the graph for x^100 looks a little off, isn't it supposed to be much steeper?

[@Mathacy]

Thank you for the kind words Amrish. As to the graph of x^100 and e^x, if x>648 then e^x > x^100. The main point of of that part is to show how dominant exponential functions are, so you can think of the x-axis scale as being very large.

[@kayihanthescientist6881]

Really nice video. Thanks for your explanations.

[@chackken]

Nice one, great job!

[@dilipranade]

simply a wonderful explanation got it first time

[@nithinsikinam]

heavily underated channel

[@Av-fn5wx]

This channel deserves subs at the rate of e^x!!

[@ARS-fi5dp]

Thank you for the useful content 🙏

[@anupamyadav5038]

Hey, you are great teacher, dude. Thanks a ton.

[@mosab643]

What software do you use to make the animations? Or, do you code it all? Btw, great job.

[@iamtrevor1551]

Please start making more like this

[@er-s428]

This was a great video thank you so much! :D

[@car103d]

I am approaching an e^exponential headache, but it’s very interesting, thanks!

[@keftam]

A fantastic explantion of the relationships.

[@jononeill5684]

great video - thank you

[@0Christ1]

Thanks for all your expressions

[@matematicacommarcospaulo]

Excellent video... Could you please give me any hint of how I create animations with LaTeX formulas like you do?

[@VolkGreg]

Very well done

[@jsingh6931]

Really good video 👍

[@elhatimimohammad6944]

I'am a senior in high school and this helped me so much when working with these functions in math and physics

[@user-bt8ru5fr6u]

My god what an excellent video!!

[@saltstillwaters7506]

Thank you very much you're very skilled

[@vytg706]

what a great channel!

[@genesis553]

Wooow mindblowing😍😍

[@Eds0n2021Gamer]

great video man!

[@neiloppa2620]

Great video!

[@vidyadharkottisa8367]

Too good. Excellent. Thanks

[@nassimzouaoui469]

mind blowing mathematics

[@nikkibhamra7536]

Really well explained I enjoyed this video

[@msag68]

🤓

[@miguelrosales581]

Thank you a lot i have been watching lot of videos trying to understand better what is e and with your your explanation i got it in no time although im not a native english speaker🎉🎉🎉

[@nikkibhamra7536]

Brilliant 👍🏼

[@lemonmorning4204]

Wow, just wow. Its beautiful.

[@Jadem4]

Nice Video!

[@Mahdi.alkawaz]

Thank you very much amazing content

[@dylanalexander5163]

what an amazing video

[@oscarvirdee7354]

Nice Vid

[@algorithminc.8850]

Fun channel. Thanks. Subscribed. Cheers

[@comcilliak]

great video, thank you. Coming from numberphiles video of e however, i am a little confused - wouldnt it be eulers number and not eulers constant? Which is supposed to be something slightly different?

[@Mg_887]

It's just 2 ways of saying it

[@nicholas6094]

Amazing video

[@ThreeDogKnightAZ]

Thanks for the explanation, Squidward!

[@sukhibhamra5750]

Really👌🏻 clever 👏🏻👏🏻

[@yassinenajar4369]

This has the same feeling as discovering lost lore that's filled with plot twists

[@LepusCompositions]

amazing video!

[@joerob5917]

This is beautiful

[@mohitgehlot6582]

Thank you so much

[@josephhuelbig]

Energy can equate to Mass dissolving multipled by constant multipled by imagery vapor symbol representation squared E≠imc (-)E=(imc)² E≠(imc)² (-)E=imc² E≠imc² (-)E=((i)(mc))² E≠((i)(mc))² (-)E=mic² E≠mic² E=(mc/2πr)² E≠(imc/2πr)² E=mc² E=i In mathematics and physics, the symbol "c" can represent various concepts and constants, including: 1. Speed of light: In physics, "c" typically represents the speed of light in vacuum, approximately \(3 \times 10^8\) meters per second. 2. Constant: "c" can also represent any constant value used in equations, formulas, or expressions. 3. Circumference: In geometry, "c" often represents the circumference of a circle, given by \(2\pi r\) where \(r\) is the radius. 4. Center: In geometry or coordinate systems, "c" can denote the center point of a circle or other geometric figure. 5. Complex number: In complex analysis, "c" is sometimes used as a constant in complex numbers, such as in expressions like \(z = a + bi\), where \(c\) represents the real part of a complex number. These are some of the common uses of the symbol "c" in mathematics and physics, though its specific meaning can vary depending on the context.

[@MohamedAhmed-gh6cs]

Thank you very much.

[@WarzoneMasters]

amazing thanks !!

[@Sara-gw5hv]

Thank you please make more

[@nandakumarcheiro]

The graph of y=x swings between exponential power of e upto x and by inversion becomes a logarithm of x.By Lambert function this becomes a complex number w multiplied by e to the power w.This means a complex may swing between an exponential by the power w as well may become an inverse function as logarithm of a complex number.

[@kristianmorris9738]

Excellent.

[@ArianTheRestless]

Very good video

[@tradernfts1377]

That was quite useful

[@kailashsingh9737]

Very nice sir

[@YorangeJuice]

5:44 that was cool i've never seen that derivation before

[@kaholeung9443]

Thank your for your tutition

[@arjun_12]

U thought us calculus in the simplest way possible than any schools or coaching institutes did.

[@Oscar-vd4cv]

Seems like he uses logarithms to define logarithms. I'm sure this video could be useful, but I'd have to watch it over and over and slow the speed.

[@davidlixenberg5999]

I found the video much too fast and when all the different k's were exchanged for different values along the grraph of e questionable. David Lixenberg

[@SpaceWoods]

@@davidlixenberg5999bro really signed off a youtube comment with his name

[@jimsvideos7201]

In the case of radioactive decay the mass isn't lost so dramatically, what changes according to that function is the concentration of that particular radioisotope. They all end up as Pb, some just take more steps to get there.

[@Zackarius]

I will worship the exponential function right now, holy shift.

[@rektspresso7288]

OH MY GOD, this at the beginning of the video is what I was trying to understand. I'm shocked

[@neerajrameshshiledar8131]

Amazing

[@neuu7171]

que videazo

[@mecoderstudios7607]

the lore got me good =D

[@vanskis7618]

HOLY SHITT!!! my mind blew

[@captainsk]

We never taughtt the derivatiion of thiis value. Thanks.