Solving x^x=2 (exact answer)

[Nxnxxx]

Here before Riemann hypothesis is solved.

[Rundas]

An easier solution in my opinion would be the following:
use ln to get x*ln(x)=ln(2)
rewrite x as e^ln(x) then apply w(x) to get ln(x)=w(ln(2))
and exponentiate.
I think that would be easier. But I can be wrong. Nevertheless, great video and greetings from germany.

[SlideRulePirate]

Computer? I used the Log-Log scales on my British Thornton AA010 Comprehensive slide rule.
I got 1.56.

[youknowwho]

Use a calculator to keep getting close

[michel_dutch]

We need more LambertW videos. That function is just cool.

[debadityabhattacharya]

You know, this video gives me the satisfaction that out there there are people who care about finding the solution to questions like these, for I too often get mocked for asking such things in class.

[TomJacobW]

Down-voted for clickbait - those two fish aren't alike! That base-fish looks like a cheesy x-prime to me! /s

[guepardiez]

Pro tip: the quickest way to solve this is by entering different values of x on a calculator and seeing which one gives x^x = 2.

[adandap]

Very neat, but really the solution is "exact" in the sense that a function is defined to be exactly the solution of this sort of problem.

[gredangeo]

Wait a minute. That pen is green. *GASP!*


What is the world coming too?? ;P

[sarishtagupta]

When you are in hurry
1¹= 1
2²=4
So the number is between these
Though 1 is more close to 2 than 4 but this is exponential growth so the answer should be near the arithmetic mean of 1and 2 that is 1.5
I used calculator to verify this and got that the answer is roughly equal to 1.56
Solved in 20 seconds ...

[omarifady]

There is an amazing method to calculate the w function by newton's method in the calculator!
To calculate W(x), in the calculator enter whatever number and press =
So this number will be save into "Ans"
And then enter this :

And then press = too many times, you will get a very very nice approximation of W(x)!

[i_am_anxious]

I solved x^x=2 with only ln(...)—>
x^x=2, so x^(2x)=4. And 2=x^x, so we substitute; 4=x^((x^x)x)=x^(x^(x+1)).
Now take the ln of this;
ln(4)=ln(x)(x^(x+1)). Divide by ln(x), and include the ln(x) in the ln(...) on the other side; ln(4-x)=x^(x+1). Now take the ln(...) again—> (x+1)ln(x)=ln(ln(4-x))
Divide by ln(x)—>x+1=ln(ln(4-x)-x). Raise e to the power of both sides—>
e^(x+1)=ln(4-x)-x. Put in in the division form again—> e^(x+1)=(ln(4)/ln(x))-x
Now, ln(2)=ln(x^x)=xln(x). Where can we get this? Including the “-x” in the fraction—> e^(x+1)=(ln(4)-xln(x))/ln(x). Now substitute xln(x) for ln(2), and ln(4) for 2ln(2)—>
e^(x+1)=(2ln(2)-ln(2))/ln(x).
2ln(2)-ln(2)=ln(2), and ln(x)=(1/x)(ln(2). Apply these—>
e^(x+1)=ln(2)/((1/x)(ln(1)). Cancel the ln(2)’s out—> e^(x+1)=1/(1/x)=x. So, e^(x+1)=x. Take the ln(...) of both sides—> x+1=ln(x). We had earlier x^(x+1)=ln(4)/ln(x), so substitute.
—>x^(x+1)=ln(4)/(x+1). Take the ln(...) of both sides;
(x+1)ln(x)=ln(ln(4)/(x+1)). We just confirmed 1+x=ln(x), subsitute—>
(1+x)^2=ln(ln(4)/(x+1)). Let’s look at (x+1)^2 for a minute.Thats x^2+2x+1=x^2+x+(x+1)=

=ln(2x). This was originally
ln(ln(4)/(x+1)), so raise e to both sides; 2x=ln(4)/(x+1). Multiply by (x+1)
—>2x(x+1)=ln(4). Now we’re talking. 2x^2+2x=ln(4), so 2x^2+2x-ln(4)=0. Now solve the quadratic. I used the quadratic formula, and I got
(-1+/-sqrt(1+ln(16))/2. Now Someone just has to find which it is. Wait, I did that using common knowledge; if it’s the negative solution, it’s a negative number to a negative power. The number is then (1/(-m))^(something)=
-n. But 2 is positive. Therefore, if x^x=2, x=(-1+Sqrt(1+ln(16))/2.

[stevenschilizzi]

Blackpenredpen, your videos are truly brilliant! I hope you can share your enthusiasm and skill in making things clear also as a $-earning job, whether in a school or a University or elsewhere. It’s great how you make the choice of solution strategy clear. The trick of course is: How does one figure out the solution STRATEGY? - What your videos seem to suggest is that math is 25% knowledge (of theorems, properties, etc.), 25% disciplined logic, and 50% creative imagination! - the 50% they never teach you in school...

[Gold]

This is no better than saying G(x) is defined as the inverse of x^x, and the answer is just G(2).

[hungryfareasternslav]

I did it in a Planck Time!!!
x^x=2
²x=2
x=ssrt(2)
where ssrt(x) is super square root of x and ²(ssrt(x)) = ssrt(x)^ssrt(x) = x
Easy!!!

[Myactualyoutubechannel]

2:36 I love how he uses 2 colors at the same time

[brennolavigne]

I don't know if u guys have payed much attention, but this "proof" means absolutely nothing. It's merely walking around in circles. The end result expresses x in terms of a function that is defined to be the inverse of xe^x, which we then use in order to calculate x, (w(ln2)). But in order for us to compute this result, for which there is no exact solution, we have to resort to numerical methods of calculation, which we could've used to approximate x^x=2 to begin with!

[ananyapathak]

Can you Please proof that formula of infinite nested root with alternate + - sign. In that x²-5=√(5-x) video.

[blue_blue-]

There is a calculator which has the W-function (and a lot more). It is the WP 34 s, a program you can flash to an HP 20 b or 30 b calculator (but difficult because missing cables.) Or use it on apple devices. Or buy it on commerce.hpcalc.org.
Emulator for mac, windows and linux on sourceforge available.