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

  • Thread starter Benn
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  • #1
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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.
 

Answers and Replies

  • #2
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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.
 
  • #3
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okays... thanks a lot!
 
  • #4
dextercioby
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Your equation has a solution, because both functions are monotone and increasing.

[tex] 3^{-1}<(-1)^2 [/tex]

[tex] 3^0 > 0^2 [/tex]
 
  • #5
D H
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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
[tex]x=-\sqrt{3^x}[/tex]
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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