Question

The value of ${\log }_{a}N\cdot {\log }_{b}N+{\log }_{b}N\cdot {\log }_{c}N+{\log }_{c}N\cdot {\log }_{a}N$ is equal to

• A. $\frac{{\log }_{a}N\cdot {\log }_{b}N\cdot {\log }_{c}N}{{\log }_{ac}N}$
• B. $\frac{{\log }_{a}N\cdot {\log }_{b}N\cdot {\log }_{c}N}{{\log }_{ab}N}$
• C. $\frac{{\log }_{a}N\cdot {\log }_{b}N\cdot {\log }_{c}N}{{\log }_{bc}N}$
• D. $\frac{{\log }_{a}N\cdot {\log }_{b}N\cdot {\log }_{c}N}{{\log }_{abc}N}$

Answer 1

The correct option is D $\frac{{\log }_{a}N\cdot {\log }_{b}N\cdot {\log }_{c}N}{{\log }_{abc}N}$

${\log }_{a}N\cdot {\log }_{b}N+{\log }_{b}N\cdot {\log }_{c}N+{\log }_{c}N\cdot {\log }_{a}N$
$=$$\frac{\log N\cdot \log N}{\log \left(a\right)\cdot \log \left(b\right)}+\frac{\log N\cdot \log N}{\log \left(b\right)\cdot \log \left(c\right)}+\frac{\log N\cdot \log N}{\log \left(c\right)\cdot \log \left(a\right)}$
$=$$\frac{(\log N\cdot \log N)(\log a+\log b+\log c)}{\log \left(a\right)\log \left(b\right)\log \left(c\right)}$
$=$$\frac{(\log N\cdot \log N)\left(log\right(abc\left)\right)}{\log \left(a\right)\cdot \log \left(b\right)\cdot \log \left(c\right)}$
Multiplying Numerator and Denominator by $\log N$
$=$$\frac{(\log N\cdot \log N\cdot \log N)\left(log\right(abc\left)\right)}{\log N\cdot \log \left(a\right)\cdot \log \left(b\right)\cdot \log \left(c\right)}$
$=$$\frac{(\log N\cdot \log N\cdot \log N)}{\log \left(a\right)\cdot \log \left(b\right)\cdot \log \left(c\right)}\times \frac{log\left(abc\right)}{\log N}$
$=$$\frac{{\log }_{a}N\cdot {\log }_{b}N\cdot {\log }_{c}N}{{\log }_{abc}N}$