Solve for x (3^x = x^2)

[Benn]

1. The problem statement, all variables and given/known data

Find all possible values of x:

x2 = 3x

2. Relevant equations



3. 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.