Question

The value of $[\frac{1}{{\log }_{\left(p/q\right)}x}+\frac{1}{{\log }_{\left(q/r\right)}x}+\frac{1}{{\log }_{\left(r/p\right)}x}]$

• A. $0$
• B. $1$
• C. $2$
• D. $Noneofthese$

Answer 1

The correct option is A $0$

We have,

$\begin{array}{c}\frac{1}{\log {\left(\frac{p}{q}\right)}^x}+\frac{1}{\log {\left(\frac{q}{r}\right)}^x}+\frac{1}{\log {\left(\frac{r}{p}\right)}^x}\\ \Rightarrow \log {\left(\frac{p}{q}\right)}^{-x}+\log {\left(\frac{q}{r}\right)}^{-x}+\log {\left(\frac{r}{p}\right)}^{-x}\\ -x\left(\log p-\log q\right)+\left(-x\right)\left(\log q-\log r\right)+\left(-x\right)\left(\log r-\log p\right)\\ =-x\log p+x\log q-x\log q+x\log r-x\log r+x\log p\\ =0\end{array}$

Hence, this is the answer.