Need help solving 2^x = 1/x

not a simple one
the easiest way would be to graph y = 2^x, and y=1/x and look for the intersection, its about 0.64

crapppp this is hard haha. cant find a way (besides using calculator, which is basically the super easy way)

yeah I estimated it with my calculator, but I'm wondering if there's a way to actually work it out. I have no idea... I'm wondering if there's a technique I could use to find the inverse of x*2^x and plug in 1 for the solution... but so far I can't invert it

heh, let it go man. There is no easy way to get this one other than asking a computer which is just going to:

try 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 then realize the answer is between 0.6 and 0.7
try 0.61, 0.62, 0.63, 0.64, 0.65 then realize the answer is between 0.64 and 0.65
try 0.641, 0.642, ...

until it gives up or you tell it to stop (which is probably similar to what you did, it can just do it much faster than you).

This post was edited by Azrad on Aug 25 2013 10:25pm

Impossible to solve in an algebraic way.
Only iterative calculus can give an approximate answer - as said above.

However, it is possible to prove the existence and unicity of a real root.

there must be a real solution, but 'x' is transcendental

surely x<1, so now
first prove that x is not rational, which is quite easy to see because assuming x=p/q would result in: 2^(p/q) = 1/(p/q) = 2^p 2^(1/q) =>

p . 2^p . 2^(1/q) = q, now 2^(1/q) is irrational for any integer greater 1

with x being irrational is follows from gelfond-schneider that 2^x is transcendental and not being an algebraic number there is no finite polynomial solution
if there is some 'simple' series to represent it, i am not going to try to work it out - have fun, try it yourself