solve e ^x + 2x=7?

Question

math wiz anyone?

Answer 1

As the other posters have mentioned, this has no closed-form solution in terms of elementary functions. If, however, you have access to the Lambert W function, you can obtain a closed-form solution as follows:

First, exponentiate both sides:

e^(e^x)e^(2x)=e^7

Take the square root of both sides:

e^(e^x/2)e^x = e^(7/2)

Divide both sides by 2:

e^(e^x/2)e^x/2 = e^(7/2)/2

Invoke the Lambert W function:

e^x/2 = W(e^(7/2)/2)

Multiply both sides by 2:

e^x = 2W(e^(7/2)/2)

Take the natural log of both sides:

x = ln (2W(e^(7/2)/2))

And we are done. This is, as the other posters have mentioned, approximately 1.42372338

Answer 2

It's a trascendental equation, it has no closed solution. You can solve it by numeric methods like Newton-Raphson.

Oh, if you were just looking for the answer, it's aprox. 1.42372 (done with Mathematica 4)

Edit: Oops, good addition by Pascal, I forgot to mention that there's no close solution "in terms of elementary functions" (I'm using " " to enphasize, because I can't think of anything better).
Also, I didn't know about the Lambert function, thanks for sharing it Pascal.

Answer 3

1.4236