solution to the logarithmic triangle

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Here's a fun math problem when we have different logarithms on a right triangle. Is it possible to solve for x so that ln(x), ln(2x), and ln(3x) form a legitimate right triangle? Of course, we will need to use the Pythagorean Theorem and get a triple logarithm equation. Then we will also be using many of the logarithm properties and the quadratic formula to solve this problem. Subscribe to @blackpenredpen for more fun math videos!
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19 июл 2022

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Комментарии : 390

@blackpenredpen Год назад

Click here to check out Brilliant 👉 brilliant.org/blackpenredpen/ (20% off with this link!)

@lapicethelilsusboy491 Год назад

I'll ask again: Are you okay?

@leonardobarrera2816 Год назад

Thanks for the video

@jamespat7975 Год назад

How to solve this integral question ? Integral [ ( 1/x^2 * (1+x^4)^0.5) ] dx ?

@Abdul-ot3lu Год назад

Is brilliant worh

@hustler3of4culture3 Год назад

Brilliant

@louisvictor3473 Год назад

Log triangles are naturally very hard to manipulate, on account of their large size and weight.

@fasebingterfe6354 Год назад

I agree

@U014B Год назад

I never wood've thought about that.

@theabyss5647 Год назад

I think it's not because they're heavy but because of their temperature. We're taking about an ln triangle and liquid nitrogen is not mechanically problematic.

@davidbrisbane7206 Год назад

Log triangles are good for the environment. They trap a lot of CO2 in them.

@chitlitlah Год назад

"naturally" But natural logs are a bit easier to work with than other logs.

@bartekabuz855 Год назад

Fun fact: If you try the same thing with sine you will get x=pi/6 and with cosine x=pi/4

@joaomatos6598 Год назад

How?

@oom_boudewijns6920 Год назад

Probs u use trig. Identities

@AlchemistOfNirnroot Год назад

For a sin(x), sin(2x) and sin(3x) triangle and then you got the cos(x) solution as a result of the trig identity?

@astha_yadav Год назад

If u differentiate at the right triangle square law thing, removing the powers, then raise from e as powers removing ln , then solve the quad eqn u get 3/2 which is a correct soln Edit: actually there is some thing wrong with this method though i haven't figured out what I accidentally checked for lnx + ln2x = ln3x rather than the square form, so the soln is wrong Not deleting incase someone wishes to help out

@oom_boudewijns6920 Год назад

@@astha_yadav who asked🤣he asks about the trig version

@mathmathician8250 Год назад

You should change the side length to ln(3x), ln(4x) and ln(5x) to make people to remind of the famous 3-4-5 right angled triangle. :)

@artsmith1347 Год назад

WolframAlpha gives two solutions for log^2(3 x) + log^2(4 x) = log^2(5 x) x≈0.25848 x≈0.67166

@thexoxob9448 9 месяцев назад

The 0.25 solution doesn't work because 3 times that is less than 1.. which means length is negative, which is impossible

@MasterofNoobs69 8 месяцев назад

@@thexoxob9448it is possible with complex numbers, and you are then squaring it to make it real. The i-1-0 triangle is an example of absurd triangles you can create like this. The math works out, even if the geometry doesn’t.

@zzciobzz2963 6 месяцев назад

@@thexoxob9448less than 1 isn't negative. it's between 0 and 1

@upholdjustice372 Год назад

The "Fading In" Intro is so much better!

@blackpenredpen Год назад

Thanks!!!

@CaradhrasAiguo49 Год назад

5:12 the nice little detail about completing the square here is if you do NOT simplify ln(2) - ln(3) to ln(2/3), but add [ln(2) - ln(3)]^2 to [ln(3)]^2 - [ln(2)]^2 on the RHS, the [ln(2)]^2 will CANCEL 10:00 an approximation is x = 3.8549

@Jack_Callcott_AU Год назад

@CaradhrasAiguo49 I agree, that is the number I got! 👍

@Rex-xj4dj Год назад

I did that but still got about 2.45

@abhishankpaul 7 месяцев назад

I agree with your result

@pietrofubini7833 Год назад

I finally managed to get to the solution of the problem all by my self I feel so proud, it is all thanks to your videos

@davidbrisbane7206 Год назад

I only feel relieved when I solve maths problems.

@petek1365 Год назад

I started working this out myself until I reached the quadratic in LnX at which point I realized there was a much easier way to find the solution. All I had to do was watch the video and bprp would work it out for me :)

@joebrinson5040 Год назад

BPRP, you are my favorite math teacher. Thanks for another video.

@blackpenredpen Год назад

Thank u!

@dimitrisg63 Год назад

great video! I have been watching you since 2018 and your content is constantly getting better! good job mr. bprp.

@dimitrispapadakis2122 Год назад

Είμαστε συνονόματοι και έχουμε την ίδια εικόνα προφίλ :)

@junaidhasrat11 Год назад

@@dimitrispapadakis2122 don't tell me this is your alt account

@blackpenredpen Год назад

Thank you!

@dimitrisg63 Год назад

@@junaidhasrat11 no hahaha

@otiswebb5783 Год назад

Thanks for this vid. I solved something similar inspired by this problem: instead the sides of the triangle were cosh x, cosh(2x) and cosh(3x). Took a lot of algebraic manipulation but the final answer was pretty cool. Maybe another video?

@otiswebb5783 Год назад

There are 2 real solutions for x

@Razhy04 Год назад

This x is actually a solution to log(x)^2 + log(2x)^2 = log(3x)^2 for any log base greater than one. Bases between one and zero satisfy the equasion but they don't make a right triangle as log(x) would be negative. The other solution of the quadratic formula will give the right answer for bases between one and zero.

@procerpat9223 Год назад

this is a beautiful problem, your presentation is so impeccable I have watched it several times🙋🏻‍♂️

@DrLiangMath Год назад

Wow, wonderful topic and excellent presentation!

@BrijeshsChannel Год назад

I've started watching your vids since a month and the way u explain is so cool. i could understand understand calculus at the age of 14 thanks to you! #yay

@blackpenredpen Год назад

Glad to hear 😃

@eckhardtdom Год назад

0:00 Bro came from imaginary world to real world

@Kulshummaam Месяц назад

😂😂😂

@MrShad Год назад

Your videos are amazing!!

@jens5573 Год назад

I used to hate math, but this guy has somehow an interesting way of explaining things, so I somehow just got hooked lol 😂

@GammaFZ Год назад

same, he’s the reason I’m obsessed with math too

@racool911 Год назад

This was a really good log rule refresher lol

@samocali Год назад

I love these videos

@82rah Год назад

At 9:09 you discard the negative sqrt. But this leads to a positive value of x: (3/2) exp( -sqrt( ln(3/2) ln(9) ) ) = .536676; (3/2) exp(+-sqrt( ln(3/2) ln(9) ) ) = 3.854877

@shadowgamer6383 Год назад

Even though it's a positive value of x, the side length of the triangle which is ln x will become negative. And we can't have triangle with negative sides

@shreyaschaturvedi8851 11 месяцев назад

@@shadowgamer6383exactly

@e.s.r5809 Год назад

It's simply fascinating how the quadratic formula pops up like this. More than once a non-scientist/engineer/mathematician has said to me, "They made us memorise the quadratic formula in school. Why? Where will that ever be relevant?" And the answer is... well, everywhere! If you could pick only one formula to memorise, I think this would be a strong choice!

@Someniatko Год назад

It's even better to understand how to derive this formula! It's pretty easy!

@cristianrdz7667 Год назад

@@Someniatko Yeah, is easy

@Bts.121_4 Год назад

You are brilliant👍 ☺ I love your dedication

@tambuwalmathsclass Год назад

Amazing creativity

@SG-lh7up Год назад

I saw your great older video on x^x. Would you consider making a video on plotting x^x (in 3 dimensions) for Real input and complex output? I tried to sketch the full 3d curve, with the x axis being Real and running perpendicular to the complex plane which is used for the output of x^x. So the x axis is the Real input; the y axis is the Real output and the z axis is the imaginary output. So the y and z axes form the complex plane output of the Real x input. So you have a simple exponential-looking 2d curve for positive x, it crosses the real y axis (or has a limit at x=0) at y=1, but the curve then becomes a complex shrinking spiralling "vase" shape for negative x. It's the "smoothness" of the curve as it crosses the y axis and changes from Real 2d to Complex 3d that I can't visualize. Would you consider making a video on this 3d graph and discuss the 3d smoothness of the real-complex transition at x=0 ? i.e. what's the limit of the 3d angle of the complex curve at x=0. AND: on this graph is x^x at x=0, a forbidden indeterminate point or is it equal to 1 ?

@emperorhirodripo5863 6 месяцев назад

This video was soo satisfying, because I always realised what he was about to do, split seconds before he actually did it

@itzmrinyy7484 9 дней назад

This is actually one I was able to solve by myself! Very cool, I had to attempt a variety of different methods before thinking to expand ln²2x into (ln2 + lnx)², but once I did that everything was clear.

@Goldslate73 Год назад

Please please please please please do another marathon session. Really need it. Calculus. Maybe Laplace, Fourier, Bessel etc. Please?

@antonyqueen6512 Год назад

Just a tip for quadratic equations: use simplified form of the solutions when coefficient of the linear term is even as it was the case here, i.e,: ax + 2bx + c =0 => x= [-b +|- sqrt(b^2 - ac)]/a 😉 With a=1, even simpler x= - b +|- sqrt(b^2 - c)

@anastasissfyrides2919 Год назад

Much more preferable to divide by the common factor than memorizing yet another formula

@kangalio Год назад

i know it as x²+px+q => -p/2±sqrt((p/2)²-q)

@antonyqueen6512 Год назад

@@anastasissfyrides2919 it’s not memorising new formula, it’s simplifying the 2’s

@NoNameAtAll2 Год назад

- b/2 you forgot to divide b by 2

@antonyqueen6512 Год назад

@@NoNameAtAll2 no I didn’t. That’s the whole point. It is simplified. You don’t have the division by 2. The coefficient of at x is even: 2b, thus -2b/2a= -b/a and sqrt[(2b)^2 - 4ac]/2a= sqrt[b^2 - ac]/a With the coefficient a of x^2 being a=1 you have the simplified solution as indicated in the comment above ☝️

@milmi__9582 Год назад

Thank you

@manavrana225 Год назад

Note: x needs to be greater than 1.5 as sum of two sides need to be greater than third side or the difference between 2 sides needs to be less than the third side.

@computernerd1101 Год назад

The approximate value of x is 3.85488

@Smosh7i 4 месяца назад

What about x = 0.583676

@computernerd1101 4 месяца назад

@@Smosh7i That does work algebraically, but if x < 1, then ln(x) < 0. Geometrically, it doesn't make sense for the edge of a triangle to have a negative length.

@kent631420 Год назад

Dear bprp, I have a question, and I'd appreciate it if you solve it in your next video: Find the max/min value for sinA*sinB*sinC where A, B, and C are three angles in a triangle (A+B+C=pi) Thank you

@simonwillover4175 Год назад

Picks complex A, B, C

@bebizambi392 Год назад

Possible solution?: since A, B and C are angles which form a triangle, you could take C=π-(A+B). Then, sinC= sin(A+B) due to allied angles. Resulting expression is sinA*sinB*sin(A+B). I used maxima and minima for above expression using partial derivatives and got the answer.

@davidp4427 Год назад

Help me out here. A + B + C = 180° so 180° = pi ??? Am I missing something?

@nguyenphungdunganh3941 Год назад

@@davidp4427 radians since we're adults now

@SuperYoonHo Год назад

Thanks

@DynestiGTI Год назад

I love how you just pop into existence in the beginning

@saujanyapoudel8910 Год назад

In my take, I factored out the 4 from the square root resulting in the product of the square root and 2 then I factored out 2 in the numerator and cancelled it with the 2 in the denominator. When you didn't do the same I expected you would have some twist so I was afraid if I have to rewrite it again.

@BlastinRope Год назад

Tbh in calc 2 it wasnt the calc that got me but the occaisonal algebra trick

@HebertMusingarimi-jw4wj Год назад

Well educative

@kalmes Год назад

That was actually a pretty fun problem.

@mcgyverlouw8881 Год назад

Great stuff here. When I saw the thumbnail my first thought was IS THIS POSSIBLE? Any other type of functions we can use for the sides of the right angled triangle? What about e^x?

@oenrn Год назад

He did e^x in another video.

@Peter_1986 Год назад

blackpenredpen always comes up with interesting problems.

@chazzbunn7811 Год назад

I got the same answer, I wanted to check it before watching this video. Checking it with algebra by putting the solution back into the original equation proved difficult, much harder than the actual problem in fact.

@tomctutor Год назад

Almost the same as BPRP direct analysis, notice that: log(2x) - log(x) = log(2) log(2x) + log(x) = log(2x^2) from which the product gives the difference of squares, [log(2x)]^2 - [log(x)]^2 = log(2)log(2x^2) = log(2)[log(2)+2log(x)] ...eq(1) from the triangle pythagoras, [log(2x)]^2 + [log(x)]^2 = [log(3x)]^2 = [log(x) + log(3)]^2 ...eq(2) eq(2) - eq(1) gives, 2[log(x)]^2 = [log(x) + log(3)]^2 - log(2)[log(2)+2log(x)] a quadratic in log(x), let u = log(x), u^2 - 2[log(3/2)]u - log(6)log(3/2) = 0 solve for u using quadratic formula and your done x = e^(1/2){ 2log(3/2)+- sqrt[4[log(3/2)]^2 +4log(6)log(3/2)] } etc..

@Lucretiel Год назад

I took me a while to notice how seamlessly he was switching between red and black and now I’m extremely jealous

@barndoor1262 Год назад

Has anyone noticed the WIZARDRY at the first 3 seconds of the video?!? I haven't yet watched this but the first few seconds scared the bejezsus outta me. Why did they do that? The editor must have had a chuckle.

@AfaqueAhmed_ Год назад

0:00 Just a man coming out of the blue with a Blue pen and Red pen and a sweet Log problem for us .

@charlesbromberick4247 Год назад

nice job

@usdescartes Год назад

If you solve the generalized problem of using sides ln(nx), ln((n+1)x), and ln((n+2)x), you get: x = (n+2)/(n (n+1)) * e^sqrt(2 * ln((n+2)/n) * ln((n+2)/(n+1)))

@sssilky3317 Год назад

I knew I was wrong when the answer I got was super long, roughly 3 times longer than the one you got. I checked both of their exact values to make sure it wasn't just a different way of expressing the same value and it wasn't :(

@fabiangn8022 Год назад

Gracias.👍🏽

@papasalt8823 8 месяцев назад

I believe I messed up somewhere along my working and don't feel like restarting. But from a number theory perspective, couldn't this be solved through Euclid's formula? Often used only with integers, but it applies to the real numbers too. If a^2 + b^2 = c^2. Where: a = m^2-n^2 = lnx b = 2mn = ln2x c = m^2+n^2 = ln3x We can raise everything to the power of e. Then rearrange for x in each equation. And set 3x to be equal to the sum of each equation. (3x = e^a + e^b + e^c). I'm not sure where to go from here though, but I haven't worked through far enough to think about that section, and I'm too lazy to do it since I already mucked up once.

@tommydecarite4121 Год назад

Hi thanks for the vid have a nice day it was great

@chenshan4973 Год назад

what a incredible video..

@voidkfox9526 Год назад

You forgot to distribute the square power in the b^2 of the cuadratic formula. (2ln(2/3)^2 is 4(ln(2/3))^2, not 4ln(2/3) as you say in the video

@2012tulio Год назад

After the second step just replace lnx by u and then continue that would be easier

@reidflemingworldstoughestm1394 Год назад

Pretty cool algebra 2 problem.

@bjarnivalur6330 Год назад

I was really hoping for it to work for all X

@latestmoviesforall Год назад

you should simplify the exponential of the square root.

@josephtraverso2700 Год назад

The sudden chimpmunk voice jump scared me at 8:45

@ItsPungpond98 Год назад

Bprp's top 10 catchphrases 1. Let's do some math for fun! 2. Oh my god! Looks pretty crazy! 3. Wouldn't it be nice... 4. Don't forget the plus C! 5. Today, we have the integral of... 6. Let's go to the complex world! 7. I don't like to be at the bottom, I like to be on the top. 8. Bring this down down! 9. Don't worry, don't worry. 10. The best friend of the black pen is the red pen.

@NeedBetterLoginName Год назад

I love your videos so much! I wish as a young nerd interested in math I had such wonderful resources available to me. Unfortunately I was a young nerd is 80's America, before the internet was common let alone RU-vid and pretty much the worst time to be young nerd interested in math. :-/

@ciiil8802 Год назад

Can you do 100 Linear Algebra video

@gcewing Год назад

Now I'm wondering whether there are any "log-Pythagorean triples", i.e. integers a, b, c such that (ln a)^2 + (ln b)^2 = (ln c)^2. If there are, how would one go about finding them?

@Utesfan100 Год назад

Bonus points if you use Lambert's W function

@asmmusic6336 Год назад

Can you explain some math famous problems like the zeta function or something like that

@alikanan7011 Год назад

I like this idea

@Eichro Год назад

you know the class is gonna be good when the teacher literally teleports in the room

@davidbrisbane7206 Год назад

I thought he said, "Love triangle" 😂🤣🤣

@DokterrDanger Год назад

7:50 Best part: SHWOO!

@steventrimble2275 Год назад

ln(3x)^2 = ln(2x)^2 + ln(x)^2 Then take the derivative of both sides above and cancel common factors ln(3x) = ln(x) + ln(2x) ln(3x) = ln(2x^2) ln(3x/2x^2) = 0 ln(3/(2x)) = 0 then raise to the power of e 3/(2x) = 1 x = 3/2

@AndreyNsk89 Год назад

Applying derivatives to both sides of the equation does not produce an equivalent equation. For example equation x = x + 1 doesnt have solutions, but if you take derivatives it will become 1 = 1, i.e. it is correct for all x.

@e.s.r5809 Год назад

The other issue here (besides the one Andrey pointed out-- I think you meant square root, not derivative) is that you still have to take (ln3 + lnx)^2 = ln3ln3 + 2ln3lnx + lnxlnx. (p + q)^2 =/= p^2 + q^2 sqrt(p^2 + q^2) =/= p + q (p + q)^2 = p^2 + 2pq + q^2 By Pythagoras (substituting lnx = z, ln3 = a, ln2 = b for the sake of everyone's sanity) we expand our brackets and reach: z^2 + 2az + a^2 = 2z^2 + 2bz + b^2 Gathering, simplifying, and substituting back a and b: z^2 + ln(4/9)z + ln(6)ln(2/3) = 0 As you can see, it's a quadratic with two solutions and no common factors to cancel. If you took the exponential of both sides now you'd reach an impasse (or at least something very gross). Remember you have to take exp of the entire expression, and by log rules, those multiplying lns would end up as powers: alog(b) = log(b^a) lnxlnx = ln( x^(lnx) ) 1 = exp{ ln[ x^(ln(x)) × x^ln(4/9) × ... Grim! 😅

@lotis6441 Год назад

cant I use the power rule for logs at 2(ln2)(lnx) so that 2lnx^(ln2) => lnx^(2ln2) => lnx^(ln4)?

@manavrana225 Год назад

That 2 goes to power of not power of lnx so it wiil be (ln(x²))^(ln2)

@desiaasm Год назад

X is approximately 3,8549 and is a transcendental number!

@shivamchouhan5077 Год назад

Actually it is 3.854765

@desiaasm Год назад

@@shivamchouhan5077 Yeah I just rounded it mate

@shivamchouhan5077 Год назад

@@desiaasm But you added comma (,) instead of dot(.) So your answer was 38549

@nuclear3011 Год назад

@@shivamchouhan5077 in some countries (like Poland, where I live, for example) people use commas to mark the decimal point and use dots in big numbers e.g. 1.000.000

@shivamchouhan5077 Год назад

@@nuclear3011 Oh thanks for telling I didn't know that one, btw this can lead to calculations errors in some cases.

@mr.shgamingguy Год назад

Hypotenuse and legs are on the both side of the triangle.

@jstecher526 Год назад

What song is used during the Brilliant ad?

@Uni-Coder Год назад

What about exponential triangle problem, exp(x), exp(2x), exp(3x) ?

@davidhowe6905 Год назад

I tried this just now; first of all, thought it was impossible - then noticed my basic algebra error! I got x = 0.2406 (4 decimal places). Similar method; use Pythagoras then simplify to get quadratic in exp(2x) giving exp(2x) = (1 + sqrt(5))/2 (I think this is correct)

@zahari20 Год назад

Why don't you set y-lnx from the beginning?

@nikhilsoni2403 Год назад

Wow!! I solved it by a different method (but your method is much simpler and shorter) and got this answer x = 3^[a(a+ sqrroot2)] Where a = sqrroot {[log₃(3/2)]} I thought my answer is wrong ,but after using the calculator, I found that my answer is correct !! 🥳🥳

@klaadem Год назад

0:00 man just phased into existence to teach me math 😭

@aliexpress.official Год назад

Challenge: find x such that: log(ax)^2 + log(bx)^2=log(cx)^2 for arbitrary a,b,c

@europeankid98 Год назад

x = panda

@Liamhvet Год назад

*appears out of nowhere* *starts explaining math*

@airsurfer5498 Год назад

Yea, nice problem🥳🤔

@reubenmanzo2054 Год назад

After a very exhaustive effort, I got the following solution: x=e^{-ln(2/3) (+/-) sqrt[2ln(3/2)ln3]}

@Jkauppa Год назад

btw, thats the form of primes as logarithms of base prime exponents

@Jkauppa Год назад

but the primes are just exponent one, of one ln base

@tahabouthouri7803 Год назад

Ln9 can be 2ln3 so u simplify 2 and 1/2 and you'll get (ln3)² and you simplify the square root and ull get e^(ln3) and you simplify more and you'll finally get 9/2

@TheMrvidfreak Год назад

Wow, that's a pathological side length :D

@Dviih 6 месяцев назад

Shouldn’t the final solution be x = (3/2)+e^(sqrt(ln(3/2)*ln9)) ?

@runnow2655 5 месяцев назад

you can simplify a litle further because ln(a)ln(b) = ln(a^b) so ln(3/2)ln(9) = ln((3/2)^9) and then you can find x=3/2 * e^sqrt(ln(19683/512)), looking at that now I can actually see why you didn't but I don't wanna waste the time I spent making this comment

@jamespat7975 Год назад

How to solve this integral question ? Integral [ ( 1/x^2 * (1+x^4)^0.5) ] dx ?

@pythagoras31416 Год назад

Sir some of your videos have a unique black board which can move up and down, I have seen such type of board in Oxford Mathematics youtube channel, can you please tell me name of the type of your that board???? In my country I never saw that. I want to build one myself.. take love sir

@stephenbeck7222 Год назад

Pretty common in older university buildings with large lecture halls. I think the videos you are talking about were filmed in a classroom at Berkeley, where Dr Peyam teaches. Michael Penn might be the RU-vid guy to ask about advice for building a chalk board.

@alienbsg Год назад

I solved it by splitting ln(2x) into ln2+lnx and ln3x=ln3+lnx Then by pythagoras theorem we get (ln2+lnx)²+(lnx)²=(ln3+lnx)² (ln2)²+2(ln2(lnx)+2(lnx)²=(ln3)²+2(ln3)(lnx)+(lnx)² (ln2)²-lnsolved it by splitting ln(2x) into ln2+lnx and ln3x=ln3+lnx Then by pythagoras theorem we get (ln2+lnx)²+(lnx)²=(ln3+lnx)² (ln2)²+2(ln2(lnx)+2(lnx)²=(ln3)²+2(ln3)(lnx)+(lnx)² (2ln2lnx)-(2ln3lnx)+(lnx)²=(ln3)²-(ln2)² ln(x)(2ln2-2ln3)+lnx²=ln(3^ln(3)÷2^ln(2)) (lnx)²+ln(4/9)lnx-ln(3^ln3÷2^ln2)=0 Sub Y=lnx Y²+ln4/9Y-ln(3^ln3/2^ln2)=0 One solution is Y≈1.3493 Since Y=lnx X=e^Y=e^1.3493 X≈3.855 Other solution Y≈-0.5384 X≈e^-0.5384≈0.584

@captainkarma7374 Год назад

Please dont tell me you typed all that 💀

@alienbsg Год назад

@@captainkarma7374 was tedious but that's how proofs are lmao

@asparkdeity8717 Год назад

X=0.58 not a solution as then ln(X) < 0 for one of the triangle sides

@NoNameAtAll2 Год назад

@@asparkdeity8717 why is that a problem? if x

@asparkdeity8717 Год назад

@@NoNameAtAll2 this problem represents side lengths of triangles, all of which must be positive, yet lnX < 0 so isn’t a solution; u just need one extra sentence saying this solution should be disregarded

@Fred-yq3fs Год назад

This is not too hard. Just apply the Ln formula, solve a quadratic equation, and take the exp. A year 11 should be able to do it. Takes less than a page. Great exercise and great content.

@jacekskurkiewicz4851 Год назад

Your t-shirt made me think that the golden ratio will appear in the answer...

@Kcite Год назад

dang the intro is smooth

@blackpenredpen Год назад

Thanks!

@lucachiesura5191 Год назад

that's ok!

@alibekturashev6251 Год назад

Idea for the math for fun: calculate sin(e) This will be literally 'approximation of sine'

@NightSkyJeff Год назад

I like crazy pythagorean triple questions. I have one for you... Can you find a pythagorean triple (a, b, c) such that (1/c, 1/b, 1/a) is also a pythagorean triple?

@charlesstimler9276 Год назад

The golden ratio rules!

@whocares12372 Год назад

What is the answer plz

@endeavourer1073 Год назад

@yqisq6966 Год назад

Guys this solution works! My love triangle problem is gone, thanks to this.

@krabbediem 8 месяцев назад

I was almost asleep as you entered squirrel-mode... I'm not almost asleep anymore!

@Ian_Chia Год назад

i just expanded to get a quadratic and used calculator, its so fast

@UStrom3169 Год назад

Did he forget the + between the 3/2 and e at the final answer or did I miss sth?

@DimoDimoo Год назад

Can someone explain to why why instead of x is equal to in the quadratic formula, he did ln(x) is equal to?

@pablodv87 Год назад

Because the equation he is trying to solve is not a quadratic equation in terms of x, so x is not equal to the solution of the quadratic equation. What he actually did is replace ln(x) = y, then solve the quadratic equation for y, then replace it back.

@crustyoldfart Год назад

Neat problem - more subtle than I at first thought - which was that you would be proving an identity. The solution can be summarized as follows : put a(x) = ln( x ) ; b(x) = ln( 2*x ) ; c(x) = ln( 3*x ) ; if a(x)^2 + b(x)^2 = c(x)^2 -> x = { 3*N/2, 3/(2*N) } where N = e^y ; y = sqrt(-2*ln(2)*ln(3)+2*ln(3)^2) -> N = 2.569917715.. x = { 0.5836762755, 3.854876572 } The open question is : when are mathematicians going to admit that not only are calculators here to stay but also math software ?