Question

Question #e182a

Answer 1

Take logarithms on both sides

`x^x=2`

`x*logx=log2`

`e^(logx)*logx=log2`

`x=e^(W(log2))`

`x≈1.55961`

Where `W(x)` is the ProductLog function, defined as the inverse of

the function `f(x)=x*e^x`

Also if you plot the function you will see that there no integer solutions.

Answer 2

Please see below.

Explanation:

Approximate forms
`x = e^(W(2 i pi n + log(2)))`, `i (2 pi n - i log(2))!=0`, `n` element `Z`
`x = e^(W_(-1)(2 i pi n + log(2)))`, `i (2 pi n - i log(2))!=0`, `n` element `Z`
`x = e^(W_1(2 i pi n + log(2)))`, `i (2 pi n - i log(2))!=0`, `n` element `Z`