If 3^x=4^y=12^z ,then prove that (x+y)z=xy.

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Solution

3^x = 4^y = 12^z
take 3^x = 12^z

3 = 12^(z/x).......(1)
take 4^y =12^z

4 = 12^(z/y).......(2)

multiply equation (1) and (2).
you get 12 =12^(z/x + z/y)
equating the powers on both sides :
1 = z/x + z/y
xy/(x+y)=z

(x+y)z=xy

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