A) \[0\]
B) \[e\]
C) \[1\]
D) \[2\]
E) \[3\]
Correct Answer: C
Solution :
\[({{\log }_{b}}a{{\log }_{c}}a-{{\log }_{a}}a)+({{\log }_{a}}b{{\log }_{c}}b\]\[-{{\log }_{b}}b)+{{\log }_{a}}c.{{\log }_{b}}c-{{\log }_{c}}c)=0\] \[\Rightarrow \] \[\left( \frac{\log a}{\log b}.\frac{\log a}{\log c}-\frac{\log a}{\log a} \right)\]\[+\left( \frac{\log b}{\log a}.\frac{\log b}{\log c}-\frac{\log b}{\log b} \right)\] \[+\left( \frac{\log c}{\log a}.\frac{\log c}{\log b}-\frac{\log c}{\log c} \right)=0\] \[\Rightarrow \]\[\log a\left( \frac{{{(\log a)}^{2}}-\log b\log c}{\log a\log b\log c} \right)\] \[+\log b\left( \frac{{{(\log b)}^{2}}-\log a\log c}{\log a\log b\log c} \right)\] \[+\log c\left( \frac{{{(\log c)}^{2}}-\log a\log b}{\log a\log b\log c} \right)=0\] \[\Rightarrow \] \[{{(\log a)}^{3}}+{{(\log b)}^{3}}+{{(\log c)}^{3}}\] \[-3\log a\log b\log c=0\] \[\Rightarrow \] \[(\log a+\log b+\log c)=0\] \[\Rightarrow \] \[abc=1\]
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