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Fundamental Laws of Logarithms

If x=loga bc,...

Question

If $x = l o g_{a} (b c), y = l o g_{b} (c a), z = l o g_{c} (a b),$ then which of the following is equal to 1

A

x + y + z

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B

(1 + x)^{- 1} + (1 + y)^{- 1} + (1 + z)^{- 1}

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C

xyz

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D

None of these

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Solution

The correct option is B $(1 + x)^{- 1} + (1 + y)^{- 1} + (1 + z)^{- 1}$
$x = l o g_{a} b c \Rightarrow 1 + x = l o g_{a} a + l o g_{a} b c = l o g_{a} a b c$
$∴ (1 + x)^{- 1} = l o g_{a b c} a$
$∴ (1 + x)^{- 1} + (1 + y)^{- 1} + (1 + z)^{- 1} = l o g_{a b c} a + l o g_{a b c} c + l o g_{a b c} b$
$= l o g_{a b c} a b c = 1$

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