Given-a-b-c-real-number-and-not-equal-to-1-If-log-a-b-log-b-c-log-c-a-0-then-log-a-b-3-log-b-c-3-log-c-a-3-

Question Number 119802 by bemath last updated on 27/Oct/20

G i v e n a, b, c r e a l n u m b e r a n d n o t e q u a l t o 1 .

I f \log_{a} (b) + \log_{b} (c) + \log_{c} (a) = 0 t h e n

{(\log_{a} (b))}^{3} + {(\log_{b} (c))}^{3} + {(\log_{c} (a))}^{3} = ?

Answered by $@y@m last updated on 27/Oct/20

I f l + m + n = 0

t h e n l^{3} + m^{3} + n^{3} = 3 l m n

H e r e l = \log_{a} (b)

m = \log_{b} (c)

n = \log_{c} (a)

∴ 3 l m n = 3 \times \log_{a} (b) \times \log_{b} (c) \times \log_{c} (a) = 3

Answered by Ar Brandon last updated on 27/Oct/20

\log_{a} b + \log_{b} c + \log_{c} a = 0

\Rightarrow {(\log_{a} b + \log_{b} c)}^{3} = - {(\log_{c} a)}^{3}

\Rightarrow {(\log_{a} b)}^{3} + 3 {(\log_{a} b)}^{2} (\log_{b} c)

+ 3 (\log_{a} b) {(\log_{b} c)}^{2} + {(\log_{b} c)}^{3} = - {(\log_{c} a)}^{3}

\Rightarrow {(\log_{c} a)}^{3} + {(\log_{a} b)}^{3} + {(\log_{b} c)}^{3}

= - 3 (\log_{a} b) (\log_{b} c) [\log_{a} b + \log_{b} c]

= 3 (\log_{c} a) (\log_{a} b) (\log_{b} c)

= 3 (\frac{1}{\log_{a} c}) (\frac{\log_{a} b}{1}) (\frac{\log_{a} c}{\log_{a} b}) = 3