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

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SUMMARY

The equation 3^x = x^2 can be solved using numerical methods or the Lambert W function, as analytical solutions with logarithms are ineffective. The discussion highlights that both functions involved are monotone and increasing, ensuring the existence of solutions. A practical approach mentioned is fixed point iteration, specifically using the transformation x = -√(3^x), which converges to the solution efficiently, albeit slowly by fixed point iteration standards.

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Homework Statement



Find all possible values of x:

x2 = 3x

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.
 
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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.
 
okays... thanks a lot!
 
Your equation has a solution, because both functions are monotone and increasing.

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

3^0 > 0^2
 
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.
 

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