If vanishes where

MATHEMATICS · Single correct · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

If log⁡ab+log⁡bc+log⁡ca\log_a b + \log_b c + \log_c a vanishes where aa, bb and cc are positive reals different than unity then the value of (log⁡ab)3+(log⁡bc)3+(log⁡ca)3(\log_a b)^3 + (\log_b c)^3 + (\log_c a)^3 is (a) an odd prime (b) an even prime (c) an odd composite (d) an irrational number

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