Algebraic Solution for x + 3^x = 4

[zoxee]

how can I solve this using algebraic methods? I know the solution is x = 1 from just looking at it, but not sure how to do it algebraically:

$x + 3^x = 4$

 

[janhaa]

Maybe it's possible to use Lambert omega function

http://en.wikipedia.org/wiki/Lambert_W_function

but it's not really an algebraically method

 

[LeonhardEu]

You can set f(x) = x+3^x-4, take x=1 as a solution to f(x) = 0 and look if it is monotonic to show that it has no other solutions. Algebraically is not possible.

 

[janhaa]

how can I solve this using algebraic methods? I know the solution is x = 1 from just looking at it, but not sure how to do it algebraically:
$x + 3^x = 4$

a very strange way of writing one:

$$x=\frac{-W(3^4\ln(3))}{\ln(3)}+4=1$$