Question

can anyone prove it ??
x to the power x = 2

answer ASAP

Answer 1

xˣ = 2   given.      To find x.
I dont understand what is to be proved here.

x Log₂ x  = Log₂ 2 = 1
x = 1/Log₂ x     or,     Log₂ x = 1/x
=>  x = 2^(1/x) 

for x = 0.01,   xˣ ≈ 1
            1               1
            2              4

So  xˣ = 2, for   1 < x < 2,  xˣ increases continuously for x > 1
       x = 1.5 ,   xˣ = 1.83
             1.6     xˣ = 2.12     more than 2.
So value of x can be found successive iterations as follows:

let x = 1.5,  the find next iteration value of  x = 2^(1/x)
for x = 1.5,  next x = 2^(1/1.5) = 1.5874
        = 1.5874,                            1.5475
        = 1.5475                              1.565
            1.565                              1.5572
            1.5572                            1.5606
            1.5606                            1.5591
            1.5591                            1.5598
            1.5598                           1.5595
            1.5595                           1.5596
            1.55961                          1.55961

So answer is close to 1.55961     -- it is a non terminating irrational number.

There are iterative methods to solve such equations.  We need to do as many iterations as possible to get the answer to the required accuracy. 

Answer 2

this is answer you want