Visual proof | Why ln(ab) = ln(a) + ln(b) ?
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- Опубликовано: 13 июл 2024
- #some3
This is a math video that tries to show a visual way to think about logarithms. Those functions we all know and love ;) from highschool and math classes.
The focus in this video was to loosly "proove", without any equations, the propriety that this function have, which is (ln(ab) = ln(a) + ln(b) )).
The proof is seperated into four results, each one of them is prooven using the previous ones.
Chapters :
0:00 Introduction
3:14 Result 1
5:43 Result 2
8:08 Result 3
8:54 Result 4
9:53 Actual proof
11:51 Extra
12:24 Ending
Disclaimer: This is NOT a formal proof of the propriety!!! If you want formal proof, you can look at it in any math textbook.
I did this video indeed using the library manim that the amazing creator of the channel #3blue1brown, Grant Sanderson, created and uses in his videos, and it is my submission to the contest #SoME3 .
If i just could have managed the time better than what i did, i would have added the section about visually prooving the limite (1 + 1/n)^n converge to e (Euler number) , which i am gonna keep for the next video. Hopefully :) .
I've always had this idea of doing such a video, but the complexity and the time i would have to put in it is just huge. Luckily, I took finally the decision to start it and it wouldn't been possible to finish it without the great great help of my cousin.
Lastly, i appologize for the bad quality of the audio and for my begginer level in english, it is my third language after arabic and french. But no worries, I will try to improve my accent and vocabilaries in the next videos, I promise :).
------- So thank you for watching the video and subscribe if you want more visual proof videos,. I still have a lot to show in here.
-------- Ohh, and if you are an art fan, you can follow me on instagram, @ahmedhamoune, i post my art there.
-------- And this is my gmail address if you wanna contact me about anything : radyhahmed11@gmail.com.
#Some3 #maths #visualproof #3blue1brown #logarithm #manim
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you have video bugs and misspellings and a 2vv, but it doesn't matter as the presentation and content is top notch :) cheers
2vv?
@@9remicheck 5:45 :)
For talking about areas, you'd better take absolute value of both sides
abs(-ln(1/a)) = abs(ln(a))
The negative sign comes from integral definition of ln(x) which I believe you assumed on the process.
But nonetheless great job 💚
This is spectacular as a first video.
I’ve never seen a cooler way of understanding the ln function. Adding to your opening statements, ln doesn’t just verify this property, but as most prepa student have seen, it’s the only solution to : f(xy) = f(x) + f(y)
Thanks Hacen, I appreciate this comment!
Yeah, I remember seeing this fact in prepa :), now I see it more clearly because of how connected the property with the (1/x) curve, which is the primitive of the log function.
It's technically not the only solution to f(xy) = f(x) + f(y). That's Cauchy's logarithmic equation, which has infinitely many solutions that aren't proportional to ln(x). If it were continuous at one point, monotonic on some interval, bounded on some interval, or whose graph excludes some disc on the plane, then the only solutions would be a constant multiple of ln(x).
@@qya33Could you real quick give me one of those solutions that aren't a constant multiple of ln(x)? It would have to be discontinuous everywhere, if I recall.
@@natevanderw I don't think it possible to write down such a function (or if it is, then I'm unsure of it). You're correct that it's discontinuous everywhere. The best example I can give is taking a non-linear solution to Cauchy's arithmetic functional equation (i.e. f(x+y)=f(x)+f(y)), letting it be f(x), and then f(ln|x|) would be a solution to f(xy) = f(x) + f(y), and not a constant multiple of ln(x) (although, if we're being pedantic, it would be a multiple of ln|x|).
The proof of the existence of a non-linear solution to Cauchy's additive functional equation on Wikipedia is non-constructive, so I'm not sure of any explicit examples.
en.wikipedia.org/wiki/Cauchy%27s_functional_equation#Existence_of_nonlinear_solutions_over_the_real_numbers
@qya33 thank you!
You earned a sub from me. I'm judging in SoME, so I'll give my feedback here the same way I would there.
This is a brilliant visual proof for this property of ln. I always just associated it with it's conguency with n^(ab) = n^a × n^b, but I cane away from this video with a much greater i tuitive understanding of this property, and I always prioritize and value building intuition over memorizing a fact.
Your explaination for depicting how to show that two rectangles have equal area by using parallel lines is very clever.
I loved the visual demonstration that created the inverse function. I might have like to see a second square drawn in the positive region to show the creation of the positive side of the function, but this didn't take away from the beauty of the visual construction.
When calculus gets involved, it can be very difficult for audience members to keep engaged, since numbers and equations can get out of control very quickly. You managed to show integration to prove to areas are equal without a single integration formula shown on screen, keeping the focus on the visual beauty of your proof!
Everything flowed together very nicely. Proving one thing elegantly led to proving the next. You added on a little extra proof at the end which was just as interesting as the premise of the video.
There were a few of text based errors here and there, but they were not enough to detract from the video. I loved the visuals, the music, the proofs, and I believe anyone who struggled to understand this logarithmic property will likely leave this video with a newfound respect and understanding for it.
It's unfortunate you didn't meet the deadline in time, but I'm very excited to see your next video. A visual proof for e sounds very exciting. Keep up the great work!
Thanks for the subscription and the time you put for watching the video and writing this beautiful comment. I tried so hard to make this video meet the deadline in time but that didn’t happen sadly.
I posted it anyway for people like you to appreciate the intuition and the beauty behind the log function.
I put so much effort to oversimplify the proof, so this is why I structured it into these 4 ‘sub-proofs’. As you said, unfortunately, there was some text based errors here and there because of the lack of time and experience. But next time, I will try my best to minimise these errors :).
I think you meant n^(a+b) = n^a × n^b
Best SoME entry I've seen so far!
The introduction is a nice setup by giving a question to ponder through the video. And revisiting a well-known property, to then prove it from another perspective is at the heart of the event.
I agree with others that equating the logarithm with the area of the curve had undiscussed assumptions behind, but it didn't lessen my surprise on how all the pieces led to the main result.
I love visual proofs like this. Everything makes so much more sense when you can see it. Very well done!
math becomes easier when we visualise it. Unfortunately, that wasn't the way we've been introduced to it as students.
This video is definitely a masterpiece 👏
Outstanding - now that I've seen this (5 months after it was released), I can't wait for your next video - don't give up!
The quick video recaps are a nice touch. I approve.
And the animations are so enjoyable, that it was a real treat to get to watch them again.
I really liked your video! This is the kind of visual proofs i think students need to get a better understanding of something that is not so intuitive as you first encounter it. Keep up the good work! I also liked the animations!
This was a neat presentation. Very elegant argument clearly presented!
Such a great start! Waiting for another video ✨
Beautiful video! I love stuff like this and making animations for math is just a great use of time. Truly thank you and please make more and enjoy life while doing it. Have a good day!
Thank you! Will do! Glad you liked it.
This was great, learnt so much from this! Definitely keep going. Really intuitive and great visualisations.
The video is full of misspellings, misnumberings etc., but the proof is beautiful.
That muzak 😮
When I got done the video and scrolled down to like and suscribe, I was expecting to see a bigger channel. This is really a hidden gem, and I'm glad to have come across it! Keep up the good work, this was great!
same experience for me!
Fantastic video! I love the typos, they lighten it up a little 🤗
And the cognitive value is immens in these animations, esp for beginners! 👍
This is literally a masterpiece, keep going dear 👏👏👏
Qué bonitas animaciones te has sacado! Sigue así, este canal tuyo pinta bien
Very nice line of thought!
I have been seeking an intuitive basis for logarithms in the sense of what they were created for. Formulas can only do so much, and this is the first visual proof I’ve seen, so it helps a lot.
I subscribed and will be on the lookout for more.
Absolutely beautiful
I want more math videos like this
6:16 I love this little animation when moving formula corresponds to changing parameters
Remarkable proof!
Visual proof is so key for understanding, please keep it up
I loved this but it felt the integration step was somewhat glossed over. I accept that dln(x)/dx = 1/x but it would be nice if that could have been included in the demonstration.
Otherwise extremely beautiful, I loved watching it.
I agree. The end just kind of jumped over the area under this curve, 1/x, is equivalent to the natural log.
Natural logs are *defined* as area under the 1/x curve.
@@Diaming787 that isn’t stated in the video though. All I was saying is that should be mentioned before the last 30 seconds where it’s used to make the conclusion.
@MideoKuze You’re right I should’ve mentioned that logarithm will be defined by the integral of the inverse function, i skipped that part unintentionally.
A great simplified video 👍👍 you worth all support ❤❤
Visual Math is exciting, especially when calculus is applied.
great work!
This is a marvelous video. Euclid would be proud!
It’s funny because I wanted to include him in the video as a narrator but the time wasn’t enough for animation, next time 😅.
very brilliant video the geometric proof really was mind blowing and memorable, hats off to you man
this video does have some spelling errors, bugs, but overall the content and presentation as well as the explanation were amazing, though i would suggest you go in a bit deeper into the explanation for the younger audiences who might not understand some things such as limits or how the area of 1/x gives ln(x) but overall amazing video and worth a watch
Pretty elaborate proving stuff from the scratch, the visualizations were so impressive...
Wonderful work!
Amazing!
Awesome! Go on the good work.
A great work just continue
nah, man. This is spectacular
Need more of these
Incredible! ❤️
Wonderful geometric proof. I loved it. Simple.
Very useful presentation
Learn visually Sir.
Thank you
It´s really cool.... All that was needed was the Cavalieri principle.
Exactly!
great video!! more math content please!
beautiful
Love your video because very smooth
😁
❤❤❤ beautiful visual proof ❤❤❤ music not necessary 🙏🙏🙏 thanks for the video ❤❤❤
Bro why are maths so beautiful
+1 sub
One of the best
Nice!
Wow excelente ilustración
The best explanation of logarithms I have ever seen in my life.
❤
Ispired me a lot ❤
very cool video, especially since it's a first. i absolutely adore anyone who uses grant's animation tool but I wish I saw more newer presentation methods actually, and this is not directed towards you, but to all SoME entries in general. I'd love seeing even a pen and paper video on something like this, because it feels like all I see is this animation engine
anyways sorry for the petty comment. other than misspellings, phenomenal video
Great work
Love this, looks amazing. What software are you using?
Superb
It was the origin on kepler's study of stars, it defined the law f(xy)=f(x)+f(y). But noone know before how to calculate it thats why kepler used tables.
Cool
The video is very nice.Could you tell me your color of background?
What an amazing video!
I really appreciate the effort you put in it
There was few bugs, but it’s fine.
And the method you used is just BRILLIANT.
Thank you so much for such content and keep going! I’m Looking forward for your next video about the euler number😊
And I have a question about manim, which version you are using and how did you learned it?
And what previous knowledge in programming is needed so someone can use this library ?
Because I tried to learn it but I felt that it’s very hard for me 🫠
و أنا أيضا أعلم العربية (لكني كتبت التعليق بالانجليزية حتى يتسنى للجميع قراءة التعليق)، أنا آدم من لبنان، لي شرف رؤية عملك ❤
Thanks! I am glad you liked it. About manim, I used the latest version of manim Community. For the skills required, i think you need some basic knowledge of python and programming.
ههه وأيضا مرحبا بك يا آدم ، و شكرا على التعليق .
Very nice video. The two rectangles being the same area is shown in Euclid's Elements, Proposition I.43.
Great
Subscribed 😂
Thank you for this. One small point - music was unnecessary and made it a little difficult to hear you.
I loved your video, can you share the code for making this video?
Just to check as a prerequisite to understand the proof - do you need to know a bit of integral calculus that indefinite integral of 1/x is ln|x|?
I was unable to grasp why the parallel line from point b on the ordinate to the abscissa measured the distance/area ab. I would suggest that an insert into the video making this step clear is desirable.
Your pupil,
David Lixenberg
integral of 1/x never made this much sense, wow
The student meets the logarithm in high school math, in second-year algebra, but this proof requires a familiarity with integral calculus.
This video is great but there are quite a few places that makes me think we are running in circular reasoning, logic.
💜💜💜
Hopefully 1 day I will understand the math
U do well
Sorry, but I didn't get, how and why there was a leap from 1/x to ln(x) as so it proves the considered property of ln.
May be there is an implicit presumption of an integral from a to b of 1/xdx is equal to ln(x), it had to be clearly stated before the proof?
0:47 I Googled to see if Chimistry was a real thing I didn't know :P
0:00 foreign
Arithmetic
Geometry
Harmonise
Quadrilateral
Visuals mean level???
Wow
More videos plz
💙
exp(ln(a b)) = a b
exp(ln(a) + ln(b)) = exp(ln(a)) exp(ln(b)) = a b
Why is the area equal to ln(a) or ln(b) by definition? Did we define ln(x) to be the area under this curve? When?
the derivative of ln(x) is 1/x
@@the_green_snake4187 is this the definition of ln? I know it's true, but not by definition...
Sound quality might be improved
Nice! But I think it could be half as long without losing anything. Sometimes it is good to be as brutal as possible when editing the script and the final video.
I wonder about the lengths covered by each rectangle. What is the relationship between how much of 1/x is covered and what the area is? Also, are each of these rectangles unique on their respective sides?
video lacks definite integral definition - the area under the curve equals the definite integral
pre-calc viewers may find it hard to understand this - but - I'm not discrediting this video and I think it's brilliant
video lacks definite integral definition - the area under the curve equals the definite integral
pre-calc viewers may find it hard to understand this - but - I'm not discrediting this video and I think it's brilliant
I always wondered why nobody invented curve pattern pan, thats why the rasberry pie cost as much as a junky old pentium for DIY. I wanted to show how addition and multiplication were equal at infinity, but this is much better proof.
In my proof, some limit is reached in combination of permutation because, " NOT a grapefruit", is so meaningless, that by the design limitations of my own dimensional knowledge, eventually, like seven dimensions can rotate zero's clockwise, given 2^24, there's reached a dimensional permutation, perhaps due to my monkey brain and particles. lol! So you have a function New - > size (x), but in the grand scheme, "new" is an utter abomination, must be triangles, you will never catch me using triangles.
Please mske more videos!!!
That also could be a way to explain why the derivative of ln(x) is 1/x
Not really. See comments above.
Ok a) yes I´d be interested in a video where you show how you get to Euler´s number (and please don´t mispronounce that man´s name like 99.999% of English speaking folk does, his name is not >>iewler>oiler
😀😀😀
But how do you prove that the integral of (1/x) = lnx visually.
Also wouldn't it be faster to prove that for a rectangle formed by (1/x)dx where x=a, is b/a times larger a rectangle of the same dx formed at x=b. Then you reduce the height of the rectangles by multiplying 1/b and scaling the width by b times to keep the area the same, and then move all the rectangles rightwards so that leftmost rectangle touches x=b, since the original sum of width of all the rectangles was (a-1), after scaling the width of the rectangles by b, the total width will be b(a-1)=ba-b, the x coordinate of the rightmost rectangle will be ba-b+b=ba, since we know the area bounded by 1 to a, is same as b to ab, ln(ab)-ln(b)=ln(a), hence you get ln(a)+ln(b)=ln(ab)
Basically I'm taking every rectangle within 1/x from x=1 to x=a, stretching them and squishing them, and relocating them to the region between x=b and x=ba, basically same idea as the video, but without the parallel line thing. Although the parallel line thing is kinda cool
Also about the proof of the same area for rectangles in parallel line, one could use similar triangles to prove it, basically the ratio of the sides of the triangle is same, and since the similar triangles is "flipped", the length and width are scaled up and down by the same factor, causing the area to be same
10:03?
I had similar misgivings on first learning this.property of.logs. I wonder3d if there were other functions that could be manipulated in unusual fashions but usually fall upon trivial.rings but found a limited but interesting property of the constant phi and some unusual.properties.squaring it versus adding but it's still.quite interesting.
I wonder if.anyone else.looks.at a perpl3xing math fact and has that.moment where they kind.of savor.the.awe.and mystery behind it. And then you get that feeling that there really is no magic in the universe because you know that once it is.explained it.will.feel.too.obvious. a bit.like explaining a joke to an absurd detail and nothing in life is funny anymore.
I am by no means good at math but ive.always.wonder3d if 2 conti uous elementary functions can be equal over a range of.values. say f1(x) and f2(×) are two f7nction and if they could be equal say from -3 to +3 for values of x but unequl.outside those -3 and +3 x bounds.
Thanks you for video and read comment
I know its true due to log(ab) = log(a)+log(b) rule
Great content but the music is so annoying
. (.) · (shift + 3) u use the wrong point use this 6 · 5 not 6 . 5 it's very different
engeneering chimistry
🤗
This major did not exist when I went through engineering school.
Dear, speek lower. We can not hear in the cell phone.
Isn’t this 1/x, not ln(x)
it is ln(x), and that's what was proven
@@notwithouttext I realised like 5 mins after posting that the integral of 1/x is ln(x)