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A formal proof for x^2 = 2^x

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There are three solutions. Can you find a formal proof for all three?

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x=2 and 4.
Proof:

22 = 22

42 = (22)2 = 24.

The third solution is x = -.76666468...., found numerically, looking for closed form.

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Posted · Report post

Using natural logarithms, the equation becomes

x = 2 ln(x)
  1. x = 2 = 2 ln (2) = 2 x 1 = 2
  2. x = 4 = 2 ln (4) = 2 x 2 = 4

The third root is negative and solves

x = -2 ln (x)

Which has no algebraic solution.
Numerically, it's

x = -0.766664695 ....

Interested to find out what you mean by formal proof.

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Posted · Report post

roots are easy to find and 2 roots may be somehow proved. 3rd is difficult to prove.

Using log base e,


2 lnx = x ln 2
x/lnx = 2/ln2, By comparison, x=2 and also LHS= 4/2lnx, again compare, x=4.

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