Solve 2 x </msup> = x 2 </msup> I've been asked to solve th

slijmigrd 2022-07-12 Answered
Solve 2 x = x 2
I've been asked to solve this and I've tried a few things but I have trouble eliminating x. I first tried taking the natural log:
x ln ( 2 ) = 2 ln ( x )
ln ( 2 ) 2 = ln ( x ) x
I don't know what to do from here so I decided to try another method:
2 x = 2 log 2 ( x 2 )
x = log 2 ( x 2 )
And then I get stuck here, I'm all out of ideas. My guess is I've overlooked something simple…
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Answers (2)

SweallySnicles3
Answered 2022-07-13 Author has 21 answers
Your equation has two obvious solutions which are x = 2 and x = 4. The last solution is not rational ( x 0.766665) and cannot be obtained using simple functions. You cannot get the last root using logarithms.
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Joshua Foley
Answered 2022-07-14 Author has 3 answers
There is a special function, W 0 ( x ) that is the inverse of f ( x ) = x e x when the latter is restricted to x [ 1 , ). Using this, expressions of the form Y = X e X can be solved as X = W 0 ( Y ). You want to find the solution(s) to the equation 2 x = x 2 . Rewrite 2 x as e ln ( 2 ) x and raise each side to the power of 1 2 . We then arrive at
x = e ln ( 2 ) 2 x
Multiple both sides by ln ( 2 ) 2 e ln ( 2 ) 2 x to arrive at
ln ( 2 ) 2 x e ln ( 2 ) 2 x = ln ( 2 ) 2
Apply W 0 to both sides to get
ln ( 2 ) 2 x = W 0 ( ln ( 2 ) 2 )
Multiply through to find
x = 2 ln ( 2 ) W 0 ( ln ( 2 ) 2 )
Which is equal to 2.
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