Solving equation with power of x in it

[Froskoy]

Homework Statement

$$x^2+1=2^x$$

Homework Equations

N/A

The Attempt at a Solution

$$x^2 + 1 = 2^x$$
$$\log_2(x^2+1) = log_22^x$$
$$log_2(x^2+1)=x$$
Get stuck at this point - don't know where to go next. Please help!

With very many thanks,

Froskoy.

 

[micromass]

This equation cannot be solved analytically. There is no technique to find the correct answer. Either you

1) Guess a solution and prove that it is the correct one.
2) Use the Lambert W function to find an expression for the solution.
3) Find a solution numerically.

(1) will work here. But it remains to prove here that the guessed solutions are the only solutions.

 

[ehild]

Plotting both sides would help to find all three roots.

ehild

 

[jedishrfu]

how about using the e^x series definition:

http://upload.wikimedia.org/wikipedia/en/math/c/f/5/cf5d03795de766b013f91c4cb409bd9c.png

and replacing 2^x with (e^ln(2))^x --> e(ln(2)x)

it won't make it any easier but it puts it in a polynomial form where you can discard terms after a spell
to get an approximate answer.