Prove it! Properties of logarithms

[@leickrobinson5186]

I love when he gives himself a clean edit point, but then later goes “Hmm... Edit it? Nah... Let ‘em see it all!“ 😂😂

[@H0tinNYC]

Yes, I love how genuine he is.

[@drpeyam]

Omg I forgot to cut it out lol

[@leickrobinson5186]

@@drpeyam 😂😂😂

[@oscaroblivion6570]

I'm new to this great site and so I was scratching my head; "What's a Chen-Lu?" Going back 3 years I find Dr. Peyam's video on the Chen-Lu. Fell out of my chair laughing: It's the chain rule.

[@dr.rahulgupta7573]

Simple identity but simpler presentation wow ! DrRahul Rohtak Haryana India

[@tgx3529]

When I was a small child, we learned at school, that the definition ln x is, that this function is inverse to a^x function. But I heard that this function ln x existed a lot of years before a^x function. At the University we learnet any definition. We learnet this(it's from my notes): Lemma: Exists only one function on interval (0;infinity) with propertis f(x*y)=f(x)+f(y) and lim f(x)/(x-1) =1 where x go to 1. This function is named logaritmic function.It's continues, growing on (0;infinity) , ,Hf=R, and applies to it rule (ln x)'=1/x , x>0. Is only this integral from 1 /t where t in interval (1;x), x>0 the definition ln x in USA??

[@Noname-67]

It's not the definition, it's the property of ln x

[@semperciok]

you can define it whatever you like, you will always get same results

[@Noname-67]

@1412 I was talking about the integral in the video

[@Noname-67]

@1412 before that

[@elfabri666]

I had such a long complicated proof for ln(x^r), yours is great, thanks!

[@Magictrend101]

I have the shortest in my channel

[@marienbad2]

It has been so long since I did maths that I have forgotten most of it, so understanding this stuff is beyond me now, but I love watching because of the joyful and interesting presentation! So nice to see someone with a love for maths explain it like this, with a smile on their face!

[@aitijhyasaha2569]

Sir, really you're a genius. It was awesome. This type of video makes the concept of mathematics more clear and strong too. Love and regards...

[@jamesbentonticer4706]

He's speaks like 4 languages too. Your right genius fits to describe him.

[@michaelempeigne3519]

this definition says that 223 / 71 < pi < 22 / 7 . The circumference of any circle is greater than three times the diameter and exceeds it by a quantity less than the seventh part of the diameter but greater than ten seventy-first parts.

[@michaelempeigne3519]

NOte : it was translated from ancient greek writing.

[@speedsterh]

Thanks for reminding those proofs from high school ! I enjoyed these very much

[@studentstudy162]

Dr Peyam uses integrals Le Me: writes Ln(x) = a or x= e^a Similarly Ln(y)=b or y=e^b xy=e^a * e^b = e^(a+b) Taking Ln both sides Ln(xy) = Ln(e^(a+b))= a+b = Ln(x) + Ln(y)

[@MA-bm9jz]

You can also prove it via homorphisms,if f is a homorphism so is f^(-1)

[@jamesbentonticer4706]

Editing like that assures zero discontinuity errors. Brilliant.

[@Emre-tt2ft]

"dos equis", clase de matemáticas con un poco de español, me encanta!!!

[@ahmedgaafar5369]

I really like his brilliant derivation steps.

[@lucakoch3432]

Hey Dr.Peyam, I always wondered how you would prove the thing you mention at 6:24 and so I would really love to see a follow up video showing how it’s done, or maybe you can just give me some hints so I can try it on my own, that would be really cool 😁👌

[@bobochdbrew-j4k]

1:00 i think should be ln’(x) not (ln(x))’ because ln is a function and ln(x) is a number so its derivative should be 0

[@drpeyam]

Same thing

[@dianeweiss4562]

As a Freshman at UCLA, we used Apostle’s Calculus textbook that also started with the definition of the natural logarithm as you stated. Unfortunately, my class was an “honors” section and I while I could solve the problems, I couldn’t follow the proofs. It wasn’t until half a century later that by watching your videos that I could piece together the proofs.

[@tsunningwah3471]

Hi Dr Peyam!! Love from Hong Kong. May I ask a question ? Normally if you differentiate lnx, you have to deal with (ln(x+h)-lnx)/h, which hasn't been proven yet. Isn't that kind of circular?

[@MichaelRothwell1]

Very neat! Another way to prove (2) is ln(x)=ln(x/y*y)=ln(x/y)+ln(y).

[@gobberman09]

5:41 And last but not least for the power law... *Cosmo's RE-DO!* And last but not least for the power law...

[@aurangzeb5735]

Sir I have a challenging question for you! if y=2r+s and x=3r-s then find derivative of y with respect to x?

[@ajiwibowo8736]

I want to ask, why we use f(1)=f(x), and why other number doesnt work f(2) or f(3)

[@medsaifeddinekamoun2921]

They should all technically work, but f(1) is clearly the easiest.

[@PackSciences]

I remember these questions from a few years back. Unfortunately the teacher forgot to remind us which definition to use in the questions of the test. We had remembered that exp(x) was defined as the continuous function such that exp(0)=1 and exp(x+y)=exp(x)exp(y); and defined ln(x) as the reciprocal. In the end, many students, me included, answered "hey it's easy, just take the inverse function on both sides and you are done", which seemed oddly easy compared to the difficulty of other questions. I think I did 2) in a similar way but got stuck in 3) because I thought it was for reals and was a bit stuck and I think I didn't answer but the question was probably for integers only and didn't think of calculating the derivative.

[@chaparral82]

Very complicated proof. It follows just because it is the reverse function of exp(x) where exp(x+y)=exp(x)*exp(y). So exp is a group isomorphism from (R,+) to (R+,*) and ln is the backwards group isomorphism from (R+,*) to (R,+)

[@drpeyam]

No but this assumes you know that exp(x+y) = exp(x) exp(y)

[@chaparral82]

@@drpeyam we knew exp(x) first ;-)

[@camilocastrojimenez8612]

Genial dos x, saludo desde Colombia.

[@rafaelpinheiro857]

I know a general way to prove that log_a (MN)= log_a M+ log_a N. Let log_a M=x and log_a N=y. Therefore, M=a^x and N=a^y. Replacing these terms: log_a (MN)=log_a [(a^x)*(a^y) ]= log_a [a^(x+y)] = x+y= log_a M+ log_a N

[@nathanisbored]

i believe 'early transcendentals' textbooks would use a different definition of ln(x), since i think 'early transcendentals' just means that things like e^x and ln(x) are introduced pre-calculus, so you wouldnt be able to define it in terms of an integral. however, i think late transcendentals is more historically accurate, because lnx was originally defined as an integral and e^x was discovered later as its inverse. kinda interesting.

[@FlyingOctopus0]

The last example made me think what if we fix the x and treat r as variable. Sadly, it doesn't help, because derivative (d/dr)x^r is not simple and uses ln, so it's probably circular reasoning. However it works the other way, if you know that ln(x^r)=r*ln(x) then you can differentiate both size by r and solve for (d/dr)x^r.

[@PackSciences]

There is something more fundamental, it's that even if you prove f is constant depending on r, you don't prove that it is constant on x, so you would end up with f(x,r)=K(x), so you always need to differentiate with respect to x.

[@AbouTaim-Lille]

Put Ln XY = A , Ln X + Ln Y = B . first of all f(t) = exp t is a continuous strictly increasing function over IR, in particular it is Injective so f(x) = f(y) implies x=y . so taking exponential of each side we have : Exp A = exp (ln XY) = XY since exp is the reverse function of exp for XY >0. And Exp B = exp (lnX + lnY) = exp ln X . exp ln Y = X.Y. so Exp A = exp B. Again . since Exp is injective we have A=B #

[@SeeTv.]

I think you forgot to cut some things out, but still nice prove!

[@plugandsocket500]

Very elegant

[@aliasgharheidaritabar9128]

Bravo doc.it was wonderful

[@guill3978]

Is (ln 2)*(ln 3) a transcendental number? And what about ln(1+e) and ((ln ln 2)^2)*(ln 2)?

[@pritivarshney2128]

Amazing!

[@babajani3569]

Hi love ur vids, I had a request. Could you plz make some videos on the STEP Exam. It is the Cambridge entrance exam for High School students and the questions in it are absolutely brutal. There are 3 papers in total and they increase in difficulty from STEP 1 being the easiest and STEP 3 is the hardest. I think you will really enjoy some of the questions since in my opinion, having looked at both the JEE advance maths questions and STEP 3, I would say that STEP 3 is actually even harder than Jee so plz give it a go. But if you want the really hard ones then do STEP 3. STEP 2 AND 1 are still hard but not as hard. Also, if you want the hardest ones, even from STEP 3, then there is a mark scheme which has all the answers of the questions from the past years and towards the end of that, you can go to STEP 3 and they tell you which questions were done well or poorly. Hence which were hardest and which were easiest.

[@dougr.2398]

Ellen of Why has a friend in a nearby town of Ecks called Ellen of Ecks

[@sadface7457]

I like as mathematician. Someone can ask you to prove it and you actually can.

[@adityamanohar2564]

Best maths teacher ✌️😊

[@miguelcerna7406]

Excellent video but 2:59 I'm having trouble understanding how it is true for all x.

[@drpeyam]

If f’(x) = 0 then f is constant (by the mean value theorem), and if it is constants, its value is the same everywhere, so it’s equal to the value at 1

[@HichamBOUKHABZA13]

Please keep proving stuff like that I'd like you prove in the next video This: exp(x+y) = exp(x) × exp(y)

[@drpeyam]

I’ve done that already

[@HichamBOUKHABZA13]

@@drpeyam ok thanks

[@nilsastrup8907]

Cool way to define lnx, but I dont think it is that much of a natural way to define it, because you would have to know that the derivative of lnx is equal to 1/x, and to find that out you must use these famous log rules. Therefore I think the arguments are sircular.

[@karimkadry]

Dr Peyam, you explain a very good topics but sometimes you use abbreviations that they are not understood. In general, thank you very much

[@deedewald1707]

A slide 📏 ruler proves the rule using hardware in hardware !

[@dougr.2398]

Ln(1/y) = ln [(y)^-1] = - ln (y). Do you know someone named Ellen who lives at Why? If not, why (!) Are you always talking about Ellen of Why? ;)

[@rubus92202]

More proofs please :)

[@TrandusNinja]

Today I had a dream where Dr Peyam did a collab with pewdiepie

[@Gameboygenius]

In the collab they would explore the properties of ⌊x⌋

[@Kdd160]

Nice!

[@j_tinoco]

¡Genial!

[@richardfredlund3802]

very nice

[@hsjkdsgd]

Chen Lu, Prada Lu😀

[@moshebr-c9q]

Math is pure philosophy with many branches to be explored.😜

[@KingGisInDaHouse]

Just raise e to both sides...

[@willnewman9783]

But then you have to define what e^x is. From what I have seen, it is more common in an introductory real analysis class to define ln first, and define e^x as the inverse.

[@nournote]

I appreciate the mathematical rigour, starting by a definition. Not like the bprp strategy.

[@MustafaBirsoz]

I love you bro

[@monikaherath7505]

Is it me or has Dr Peyam looked really handsome lately? Haha

[@Ocklepod]

couldn't you just write x= e^k, y=e^l => ln(x)+ln(y)=k+l=ln(e^(k+l))=ln(e^k*e^l)=ln(xy) i didn't like using integral defintions for something that pops up out of the property that it is the inverse to exponential functions.

[@Noname-67]

I have a much simpler proof and for any base, not just base e Let just called the base a log xy= log x+log y a^log xy= a^(log x+log y) xy= a^log x × a^log y xy=xy

[@mhmdsalhab8254]

👍

[@deedewald1707]

Us lefties need to be creative long ago !

[@dr.rahulgupta7573]

Dr 3.14159........m.Thanks a log . for nice presentation. DrRahul Rohtak Haryana India

[@drpeyam]

LOL

[@mattetor6726]

d/dx (Ellen) :D

[@alaechoulli6111]

Ln ewwwww 🤢 it’s ln

[@tgx3529]

Ln is more univerzal then ln, if you meen Log z

[@tamingphysics]

👍