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If a^(x) = b , b^(y) = c, c^(z) = a, ...

If ` a^(x) = b , b^(y) = c, c^(z) = a, x = log_(b) a^(k_(1)) , y = log_(c)b^(k_(2)), z = log _(a) c^(k_(3)), ` , then find ` K_(1) K_(2) K_(3)` .

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We have ,
` a^(x) = b,b^(y) c and c^(z) = a`
` rArr a = c^(2) =(b^(y))^(z) = (a^(x))^(yz) = a^(ayz)`
` rArr xyz = 1 `
Also xyz ` rArr a = c^(2) =(b^(y))^(z) = (a^(x))^(yz) = a^(ayz)`
` rArr xyz = 1 `
Also xyz ` log _(b) a^(k_(1)) log_(c) b^(k_(2)). logc^(k_(3))`
` k_(1)k_(2) k_(3) log_(b^(a))log_(c^(b)) . log_(a^(c))= k_(1)k_(2)k_(3) = 1`
`rArr k_(1) k_(2)k_(3) = 1`
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