Question

If ${\log }_{p}q+{\log }_{q}r+{\log }_{r}p$ vanishes, where $p,q$ and $r$ are positive reals different than unity, then the value of ${\left({\log }_{p}q\right)}^3+{\left({\log }_{q}r\right)}^3+{\left({\log }_{r}p\right)}^3$ is

• A. an odd prime.
• B. an even number.
• C. an odd composite.
• D. an irrational number.

Answer 1

The correct option is A an odd prime.

Given
${\log }_{p}q+{\log }_{q}r+{\log }_{r}p=0$
$\Rightarrow {\log }_{p}q+{\log }_{q}r=-{\log }_{r}p$ ....(1)
Consider,${\left({\log }_{p}q\right)}^3+{\left({\log }_{q}r\right)}^3+{\left({\log }_{r}p\right)}^3$
$=\left[\right({\log }_{p}q+{\log }_{q}r{)}^3-3{\log }_{p}q{\log }_{q}r({\log }_{p}q+{\log }_{q}r)]+{\left({\log }_{r}p\right)}^3$
$={(-{\log }_{r}p)}^3-3{\log }_{p}q{\log }_{q}r(-{\log }_{r}p)+{\left({\log }_{r}p\right)}^3$ (by (1))
$=3{\log }_{p}q{\log }_{q}r{\log }_{r}p=3[\because {\log }_{a}b=\frac{\log b}{\log a}]$