SOLUTION: (x^y)=(y^x) ; (x^2)=(y^3) Find out the value of x and y

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: (x^y)=(y^x) ; (x^2)=(y^3) Find out the value of x and y      Log On


   

Question 1113751: (x^y)=(y^x) ; (x^2)=(y^3)
Find out the value of x and y
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
By inspection, there is a trivial answer x = y = 1.
But there is also a nontrivial answer.
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+x%5E2+=+y%5E3+ —>
+x%5Ey+=+y%5Ex+ and the above —> =

Taking logs:

—> +x+=+%283%2F2%29%5E3++=+27%2F8+ —> +y+=+9%2F4+
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Ans: ++highlight%28+x+=+27%2F8+%29+ —> +highlight%28++y+=+9%2F4+%29+
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Check:
+x%5E2+=+%2827%2F8%29%5E2+=+729%2F64+
+y%5E3+=+%289%2F4%29%5E3+=+729%2F64+++ (ok)

+x%5Ey++=++%2827%2F8%29%5E%289%2F4%29+=++%28+3%5E27+%2F+2%5E27+%29+%5E%281%2F4%29++
(ok)

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EDITED: removed statement that x=y=0 is a trivial answer, it is not (0/0 form).