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Question

If x,y,z are in H.P. prove that
log(x+z)+log(x+z2y)=2log(xz)

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Solution

Since x,y,z are in H.P. y=2xzx+z
x2y+z=x+z4xzx+z
=(x+z)24xzx+z=(xz)2x+z
log(x2y+z)=2log(xz)log(x+z)
or log(x+z)+log(x2y+z)=2log(xz)

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