If x^y=y^x (x,y in R,x,y>0,x!=0) and x^p=y^q (p,q in R/{0},p!=q), then product xy is equal to?

odcizit49o

odcizit49o

Answered question

2022-11-11

If x y = y x ( x , y R , x , y > 0 , x 0 ) and x p = y q ( p , q R / { 0 } , p q ), then product xy is equal to?
Solution for this one is ( p q ) p + q p q , but I do not understand how I am supposed to get here, I guess something with logarithms but not sure what?

Answer & Explanation

fobiosofia3ql

fobiosofia3ql

Beginner2022-11-12Added 14 answers

So x = y q / p . Thus you have y q y / p = y y q / p . As y > 0, we have q y / p = y q / p i.e. y ( q p ) / p = q / p i.e. y = ( q / p ) p / ( q p ) .
Hence x = ( q / p ) q / ( q p )
So x y = ( q / p ) ( p + q ) / ( q p ) = ( p / q ) ( p + q ) / ( p q )
Amy Bright

Amy Bright

Beginner2022-11-13Added 4 answers

HINT:
Let x p = y q = z p q
one of the values of x is = z q and y = z p
x y = y x ( z p ) z q = ( z q ) z p
p z q = q z p z q p = q p
one of the values of z is ( q p ) 1 / ( q p ) = ( p q ) 1 / ( p q )
Now x y = z p + q =

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