SOLUTION: Values of X satisfying : X ^ sqrt X = ( sqrt X ) ^ X

Question 1061653: Values of X satisfying : X ^ sqrt X = ( sqrt X ) ^ X

Answer by ikleyn(52919) (Show Source): You can put this solution on YOUR website!
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Answer. The only two values of x satisfying this equation = are x = 1 and x = 4.

In other words, the equation has two and only two solutions x = 1 and x = 4.

Proof

Let consider this equation on its domain x > 0:


 = .


Take the logarithm of both sides. You will get


 =     or


 =     or


 = 0.


It has two solution:


1.  ln(x) = 0  --->  x = 1,   or/and


2.   =   --->  (square both sides)  --->  x =   --->   = 4x  --->  x*(x-4) = 0  --->  x = 4

    (x = 0 is not in the domain for the equation (1), and is not under consideration).


So, the statement is proved.

Solved.

Plot y = and y =

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