If x, y, z are distinct positive numbers different from 1, such that -Maths 9th

If x, y, z are distinct positive numbers different from 1, such that -Maths 9th
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(d) 1logy x. logz x – logx x = \(rac{ ext{log}\,x}{ ext{log}\,y}\) . \(rac{ ext{log}\,x}{ ext{log}\,z}\) - 1 = \(rac{ ext{(log}\,x^2)}{ ext{log}\,y.\, ext{log}\,z}\) - 1Similarly, logx y.logz y – logy y = \(rac{ ext{(log}\,y^2)}{ ext{log}\,x.\, ext{log}\,z}\) - 1 andlogx z. logy z – logz z = \(rac{ ext{(log}\,z^2)}{ ext{log}\,x.\, ext{log}\,y}\) - 1∴ LHS = \(rac{ ext{(log}\,x^2)}{ ext{log}\,y.\, ext{log}\,z}\) - 1 + \(rac{ ext{(log}\,y^2)}{ ext{log}\,x.\, ext{log}\,z}\) - 1 + \(rac{ ext{(log}\,z^2)}{ ext{log}\,x.\, ext{log}\,y}\) - 1 = \(rac{( ext{log}\,x)^3+( ext{log}\,y)^3+( ext{log}\,z)^3-3 ext{log}\,x. ext{log}\,y. ext{log}\,z}{ ext{log}\,x. ext{log}\,y. ext{log}\,z}\) = 0          (given)⇒ (log x)3 + (log y)3 + (log z)3 – 3 log x. log y. log z = 0 ⇒ log x + log y + log z = 0           (if a + b + c = 0, then a3 + b3 + c3 = 3abc) ⇒ log xyz = 0 ⇒ xyz = 1.
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