How to Find the Solution to 3^x = x^2?

[Benn]

Homework Statement

Find all possible values of x:

x² = 3^x

Homework Equations

The Attempt at a Solution

I took the log of both sides.. didn't get anywhere. I'm not sure where to start.

 

[micromass]

You can't solve this analytically with logarithms. You can either guess the right answer and prove that it works, use the Lambert W function or you can use a numerical method.

 

[Benn]

okays... thanks a lot!

 

[dextercioby]

Your equation has a solution, because both functions are monotone and increasing.

$$3^{-1}<(-1)^2$$

$$3^0 > 0^2$$

 

[D H]

One simple way to find that solution is via a fixed point iteration. Taking the square root of both sides and recognizing that the solution is negative yields
$$x=-\sqrt{3^x}$$
This converges to the solution fairly quickly. Quickly by fixed point iteration standards, that is. Fixed point iteration is rarely fast.