Prove it! Properties of logarithms

In this video, I prove the properties of logarithms like ln(xy) = ln(x) + ln(y) by using the integral definition of the logarithm. Enjoy!

[leickrobinson]

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

[oscaroblivion]

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.rahulgupta]

Simple identity but simpler presentation wow ! DrRahul Rohtak Haryana India

[tgx]

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??

[elfabri]

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

[marienbad]

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!

[studentstudy]

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)

[aitijhyasaha]

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...

[michaelempeigne]

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.

[speedsterh]

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

[Emre-ttft]

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

[MA-bmjz]

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

[ahmedgaafar]

I really like his brilliant derivation steps.

[lucakoch]

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 😁👌

[gobberman]

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

[MichaelRothwell]

Very neat!
Another way to prove (2) is

[dianeweiss]

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.

[jamesbentonticer]

Editing like that assures zero discontinuity errors. Brilliant.

[rafaelpinheiro]

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

[bobochdbrew-jk]

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