x^x = 2
Solve for x

Question

x^x = 2
Solve for x

zzr0ck3r · Answer

this can't be solved with "normal math".

will.h · Answer

It will be equal to 
1.56

danjs · Answer

You can graph y=x^x and y=5, and look at the intersection point https://www.desmos.com/calculator x is about 2.129

3mar · Answer

I think the graph would show it more clear.

danjs · Answer

oh y=2 it is, sorry

3mar · Answer

Thanks, DanJS 
This is what I did.

3mar · Answer

Thank you for the medal!

kainui · Answer

You can put it into a closed form with something called the Lambert W function, which is defined as the inverse of $f(x)=xe^x$.

$$x^x=2$$$$x\ln x = \ln 2$$$$e^{\ln x}\ln x = \ln 2$$$$\ln x = W (\ln 2)$$$$x = e^{W(\ln 2)}$$$$x=\frac{\ln 2}{W(\ln 2)}$$ 
Hey if you're willing to call $\ln 2$ a number then certainly $W (\ln 2)$ is a number right? :P

will.h · Answer

Kainui what would the number be mate

kainui · Answer

$$\frac{\ln 2}{W(\ln 2)}$$ is the number

davejavous · Answer

Thank Kainui. I understand the graphical method but I should've clarified I was looking for an algebraic solution, which I know would be outside of a typical maths course. Thank you all.

kainui · Answer

Yeah if you wanna learn more there are some fun exercises here to try out: http://www.had2know.com/academics/lambert-w-function-calculator.html and if you want to solve things like infinite power towers of what you have $2=x^{x^{x^\cdots}}$ then you can check this out too: https://luckytoilet.wordpress.com/tag/product-log/