Question

If
$log_{a}(b) + log_{b}(c) \: + log_{c}(a)$
vanishes where a,b and c are positive reals different than unity then the value of

$( log_{a}(b) ){}^{3} + ( log_{b}(c) ) {}^{3} + (log_{c}(a) ) {}^{3}$
is


A) an odd prime

B) An even prime

C) An irrational no.



➡ Require full solution
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Answer 1

OK let's suppose
l
$log_{a}(b ) = p \\ log_{b}(c) = q \\ log_{c}(a) = r \\ now \: we \: know \: identity \: that \\ {(a + b + c)}^{3} = {a}^{3} + {b}^{3} + {c}^{3} + 3(a + b)(b + c)(c + a) \\ now \: using \: this \: equation \: we \: get \\ 0 = the \: \: equation \\ as \: p + q + r = 0 \\ hence \: after \: solving \: it \: we \: get \: this \: form \: of \: equation \\ 0 = { log_{a}(b) }^{3 } + { log_{b}(c) }^{3} + { log_{c}(a) }^{3} + 3( log_{a}(b) + log_{b}(c) )( log_{b}(c) + log_{c}(a) )( log_{c}(a) + log_{a}(b) ) \\ now \: \\ 0 = { log_{a}(b) }^{3 } + { log_{b}(c) }^{3} + { log_{c}(a) }^{3} + 3( - log_{c}(a) )( - log_{a}(b) )( - log_{b}(c) ) \\ now \: simplifying \: it \: we \: get \\ 0 = { log_{a}(b) }^{3 } + { log_{b}(c) }^{3} + { log_{c}(a) }^{3} + 3( - log(a) \div log(c) )( - log(b) \div log(a) )( - log(c) \div log(b) ) \\ so \: we \: get \\ 0 = { log_{a}(b) }^{3 } + { log_{b}(c) }^{3} + { log_{c}(a) }^{3} - 3 \\ so \: \\ { log_{a}(b) }^{3 } + { log_{b}(c) }^{3} + { log_{c}(a) }^{3} = 3 \\ hence \: our \: answer$ so it is an odd prime