If a, b, c and n are positive real numbrs other than unity , prove that

$l o g_{a} n . l o g_{b} n + l o g_{b} n . l o g_{c} . n + l o g_{c} n . l o g_{a} n$

$= \frac{l o g_{a} n . l o g_{b} n . l o g_{c} n}{l o g_{a b c} n}$

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Solution

We have,

LHS $= l o g_{a} n . l o g_{b} n + l o g_{b} n . l o g_{c} . n + l o g_{c} n . l o g_{a} n$

$= \frac{1}{l o g_{n} a . l o g_{n} b} + \frac{1}{l o g_{n} b . l o g_{n} c} + \frac{1}{l o g_{n} a}$

$= \frac{l o g_{n} a + l o g_{n} b + l o g_{n} c}{l o g_{n} a . l o g_{n} b . l o g_{n} c} = \frac{l o g_{n} a b c}{l o g_{n} a . l o g_{n} b . l o g_{n} c}$

$= \frac{l o g_{a} n . l o g_{b} n . l o g_{c} n}{l o g_{a b c} n}$

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