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Question
If x = loga bc, y = logb ca, z = logc ab then prove that `1/(1 + x) + 1/(1 + y) + 1/(1 + z)` = 1
Solution
Consider `1/(1 + x) = 1/(1 + log_"a""bc")`
= `1/(log_"a""a" + log_"a""bc")`
= `1/(log_"a"("abc")`
= log(abc) a ...`[because log_"m""a" = 1/log_"a""m"]`
Similarly, `1/(1 + y)` = log(abc) b
`1/(1 + z) = 1/log_(("abc"))"c"`
∴ `1/(1 + x) + 1/(1 + y) + 1/(1 + z)`
= log(abc) a + log(abc) b + log(abc) c
= log(abc) [abc]
= 1 ...[∵ logm m = 1]
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