Is e^x=ln(x) solvable? 

We will make b^x and log_b(x) tangent to each other here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-uMfOsKWryS4.html

"We are not doing real mathematics." -blackpenredpen

“We are not doing real mathematics. We are doing complex mathematics.” I need that on a shirt

The solutions for this equation were way too complex that i couldn't even imagine

Note: The following equations have the same solutions! 1. e^x=ln(x) *this video* 2. e^x=x 3. x=ln(x) 4. e^e^e^...=x *this video* 4. x=ln(ln(ln(....) This is a super nice property when you have f(x)=f^-1(x). See Mu Prime Math's video for more details: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-53lBKCBrENY.html

black pe^e^e^e^e^e^...n red pe^e^e^e^e^e^...n

Simple math: BLACK PEN RED PEN Complex math: BLACK PEN RED PEN BLUE PEN . . .

It's only through this channel that I learned of the Lambert W function and have become fascinated, wondering why it was never mentioned in college-level calculus.

i stared at the thumbnail for five minutes thinking before clicking. this strategy works

Your biggest fan from Russia! Love your videos so much, you're my best math teacher (and English too:)) since 2018, thank you!

3:21 wow in australia.

Reals: Nope..no solution to this thing.. Complex numbers: Hold my “i”s

Test: solve e^x=ln(x). me: "eeeeeeeeeeee...."

Let me guess, it involves the lambert w function

I took interest in maths after watching your videos. Stay safe love from Nepal

I am glad that you showed the Lambert W function. I have learned QFT and statistical mechanics for sometimes, but I didn't know about the W function. That will be helpful.

Once you get x = e^(e^x) Can you multiply both sides by e^x? That would give you xe^x = (e^x)e^(e^x) Then you could use the Lambert W function on both sides to get x = e^x

I loved the 🐧,the song and not to mention "we are not doing real mathematics"✌🏻😌

3:34 We are not doing Real Mathematics PANIk We are doing Complex Mathematics kALM We are doing Complex Mathematics PANIk But it's bprp kALM

What I was looking for, a video for the complex natural log. Thank you so much B&Rp!

Before: Black pen red pen. After: Here comes blue pen.

Awesome video as always!! I started watching your channel years ago and now you are really helping with my Further Math lessons (I’m from England). If I have a problem to suggest to you, where can I submit it?? Thanks again for a great video!

black pen red pen, and occasionally, blue pen

first time heard "lambert W function".

Wait... if e^e^e^e^... doesn't converge, is it valid to say x = e^e^e^e^... x = e^x ?

I miss the intro from 2017: " Black Pen Red Pen yaaaay!" Like if you agree

This is one of my favorite channels!

Watching from South Africa. You made me love math man! Keep feeding me

No comments? :( why is it unlisted?

This was helping with extending tetration to complex numbers. I remember Dmitry Kouznetsev mentioning this as fixed point of logarithm. I guess it can be found by trial and error but obviously there are other methodes.

Since e^x and ln(x) are inverse functions, as you drew. The only places where they can intersect is when the mirroring in the line y=x is on the same point. That means that the output of e^x=x=ln(x). This is a faster way to arrive at e^x = x. This always works for inverse functions.

when you reached x=e^e^x you could simply multiply e^x on both sides and then take a lambert w function and you would end up with x=e^x.anyway.it was a flossy video:)

You know it gets complex when blackpenredpen includes a blue pen.

My first thought was that if e^x = x, then ln(x) must equal x, which necessarily means that e^x = ln(x). So we can start by writing the question as e^x = x, and then use the W function.

haha. like the way you make it simple to complex to simple. and infinite problems always are insane :)) nice vid.

Ah... Finally, come back with another video. ❤️❤️

Note that bprp isn't wearing a mask, but his microphone IS - a full face..; no, a full BODY cover! Surely this puts him in compliance with Gov. Newsom's rules. Not to mention the extreme social distancing he's practicing. OK, first, replace x with z = x + iy Take exponential of both sides: z = x + iy = e^(e^z) = e^(e^(x+iy)) = e^(eˣ(cosy + i siny)) = e^(eˣcosy) e^(ieˣsiny) = e^(eˣcosy) [cos(eˣsiny) + i sin(eˣsiny)] x = e^(eˣcosy) cos(eˣsiny) y = e^(eˣcosy) sin(eˣsiny) At first glance, I don't see where you can go from there. [I also tried starting with the polar form, and taking ln of both sides, which turned out even worse.] Let's see how we can get anywhere with this. I smell the Lambert W function, somehow... Fred

The solution is re^ia where r=a/sin(a)=e^[a/tan(a)] Now this can be drawn on desmos and my god there's a billion solutions

Can you maybe so a video on how to evaluate the infinite power tower? I guess it's probably with some sort of a sequence and a difference equation, but I would like you to show us how to do that.

"pause the video, and think, about, this." "... and today we have this guy. ok so-" 😂

If you treat x as a+bi, and use the (equivalent) e^x = x equation, you can show that there are an infinite number of solutions on the complex plane. The solutions are all in Quadrant I, asymptotically approaching the curve b = e^a, where b takes on the values of pi/2 + 2*pi*n for large n.

that last sound really makes my day!

Could have done with a demonstration of why an infinite tail of exponents is allowed, when it appeared that it would have to terminate with an x power after a finite and even number of e terms.

Love from India . I greatly admire your maths skills and teaching .

Nice work. I see that if we multiply both sides by x and solve we get a. x exp(x) = x * ln(x) b. Now apply the Lambert W function to both sides to obtain x = W(x * ln(x)) c. Now if we assume we can use the following identity, W(x*ln(x)) = ln(x), so the last result becomes x = ln(x) d. We can solve x = ln(x) using the Lambert W function to -1 =- ln(x) * exp(-ln(x)) --> -ln(x) = W(-1) e. x = exp(-W(-1)) = W(-1)/(-1) = -W(-1) or x = -W(-1)

I was never thinking there are fixed points for exp() (except for transfinite numbers). Nice to know.

that hello at the beginning makes me happy:)

"hello, let's do some math for fun" ctrl+w

AMAZING!! Your skills never stop amazing me!!

It's very interesting! I don't understand English very well(?), but, the process you write on the whiteboard is very easy to understand!

Here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-elQVZLLiod4.html&ab_channel=3Blue1Brown

I have that same bob-omb plushie haha

these recursive thing is mind blowing... nice I get to learn new stuffs

This morning, I was asking myself how could I solve ln(x) = e^(x), and I find your video the evening of the same day !

So if you plotted e^x and ln(x) in a 3D graph where z is the complex axis, would it look like two sheets that just barely touch each other at 2 points (at the two complex solutions shown)? Also, is there a cleaner representation of the solutions, or are both the real and the complex components transcendental?

Just learned about log and ln in class, this blew my mind

i would really love to see how to compute that lamber w function value at the end

Best microphone!

Always enjoying Watching your videos since when I was Grade 6, Nice Video as always! Love from the Philippines! And yeah, I am now in Highschool :)

Holy crap ! I actually thought and tried to solve the equation and got the correct answer . Never thought could do it

Black pen red pen you are the best at what you do

Thanks for this video, congratulation since Mexico!!

i love the sound effects

Sir ur the best teacher of maths wish u were here in India to teach us ur teaching skills are amazing and u r the best................🙏

Lambert W function exists: *blackpenredpen : I can milk you*

when i was trying this on my own, at first i multiplied both sides by x to get xe^x=xln(x), and then took the lambert w function to get to x=ln(x)

Love your videos

could you do a video on euler fonction, I have encountered it many times but still don't understand it.

How it would be grafically with the complex answer? Thanks for the videos! Keep going!

Blackpenredpen I am genuinely interested in being an expert in algebra and number theory. I am currently in grade 12(A2). Could you please suggest me some books? I really enjoy your videos and they have really influenced me. It would be like a dream come true to take suggestion for brainy maths book(the book with sufficient examples and questions) from a great teacher like you.

You should try to solve a^x=log_a(x) where a>0. The critical value is a=e^(1/e). For a greater no solutions, for a less two solutions.

Cool video BPRP!

Having to increase the amount of e’s in the power tower all the way to infinity to be able to reduce it to a single one 🤯

Someone may have already mentioned this? One can manipulate the equation e^(e^x)= x by multiplying both sides by e^x. Then applying the Lambert W gives e^x = x. The rest of the solution is as you showed. That eliminates the need to explain e^e^e^e^.....

Very well done.

On that day, we saw blackpenredpenbluepen

I need a breakdown of the W function!

you are really good at maths

Hey man love your t shirt

0:57 does it work if the function is monotonic in general too?

e^x = ln(x) e^e^x = x substituting the equation for x you get an infinite power tower of e^e^e^..., so you can just write e^x = x bring to one side: xe^(-x) = 1 -xe^(-x) = -1 using the lambert-W function: W(-xe^(-x)) = W(-1) = -x x = -W(-1) according to wolfram alpha its approximately equal to: 0.318 - 1.337i

Hey dude! Can you do limit of (3/2)^n*((sqrt(3^(2)+5n*2^n)-sqrt(3^(2n)+4))/n) as n goes to infinity? I think it should be fun and I would like to see your approach. Thx and have fun

Awsome and clean

Fun fact for video game fans: The mascot used for microphone is from the Famicom game "Gimmick!" (released for NES as "Mr. Gimmick") made by Sunsoft. This character in this game is one of types of enemies.

A good question be find the minimum distance between e^x and ln(x) Hint-line x=y

Just a question in my mind, since the solution is complex (as expected), is there any periodicity in the stated solution?

I've literally been working on this exact problem, except it was abs( ln(x )

Thank you sir!

Is there a video on Lambert w function?

e^^infinity would have been the perfect oppurtunity to make practical use of tetration.

"No, just no." 😂 love it

You’re gonna need more pen colors if you keep doing problems like this. Love the + C shirt. Only true math geeks get that one. 🤣

5:05 i thought the fish is going to be summoned...

Is it possible that we use Euler equation for getting the solution sooner ?

Can you make video how to arrange matrix 4x4 to diagonally dominant ? I want to solve equation with gaus seidel iteraton

First multiply both sides by x and the take w lambert on both sides to get e^x=x. Next multiply both sides by e^-x and -1. Take w again and multiply both sides by -1 to get answer: x=-lambert w(-1)

3:25 PAGING DR PEYEM! PAGING DR PEYEM! YOU ARE NEEDED HERE!!!!!

Does the "tower of e" says that e^x = ln(x) if lim x-> inf ?

i don't even need to solve this, i'm just watching cause i like your videos

AAARRRRGHHHHH TENGKIYUUUU ❤️❤️❤️