If `x^[ln[y/z]].y^[ln[xz]^2].z^[ln[x/y]]=y^[4lny]` for any x>1,y>1 and z>1, then which one of the following is correct?

Options

A. `l`n y is the GM of `l`n x, `l`n x,`l`n x and `l`n z
B. `l`n y is the AM of `l`n x, `l`n x,`l`nx and `l`n z
C. `l`n y is the HM of `l`n x, `l`n x and `l`n z
D. `l`n y is the AM of `l`n, In x, `l`n z and `l`n z

Answer

B

Explanation

`x^(ln((y)/(z))).y^(ln(xz)^(2)).z^(ln((x)/(y)))=y^(4lny)`
`impliesln[x^(ln((y)/(z)))]+ln[y^(ln(xz)^(2))]+ln[z^(ln((x)/(y)))]=ln[y^(4lny)]`
`implies[ln((y)/(z))lnx]+[2ln(xz)lny]+[ln((x)/(y))lnz]=4[lny]^(2)`
`implieslnx[lny-lnz]+2lny[lnx+lnz]+lnz[lnx-lny]=4[lny]^(2)`
`implies3lnx+lnz=4lny`
`implies(lnx+lnx+lnx+lnz)/(4)=lny`
`:." lny is the AM of `lnx,lnx,lnx,lnx" & "lnz.`