Question

If $p={\log }_{245}175$ and $q={\log }_{1715}875$, then the value of $\frac{1-pq}{p-q}$ is

• A. $1$
• B. $2$
• C. $3$
• D. $5$

Answer 1

Answer: (D)

$5$

$p={\log }_{245}175$
$\Rightarrow p=\frac{\log 175}{\log 245}=\frac{\log \left(5^2\right)7}{\log \left(7^2\right)5}=\frac{\log 7+2\log 5}{2\log 7+\log 5}$
$[\because {\log }_{a}b=\frac{\log b}{\log a},\log (ab)=\log a+\log b,\log a^m=m\log a]$
$q={\log }_{1715}875$
$\Rightarrow q=\frac{\log 875}{\log 1715}=\frac{\log \left(5^3\right)7}{\log \left(7^3\right)5}=\frac{\log 7+3\log 5}{3\log 7+\log 5}$
On substituting $\log 5=\alpha$ and $\log 7=\beta$, we get
$p=\frac{2\alpha +\beta }{\alpha +2\beta }$ and $q=\frac{3\alpha +\beta }{\alpha +3\beta }$
Therefore, $\frac{1-pq}{p-q}=\frac{(\alpha +2\beta )(\alpha +3\beta )-(2\alpha +\beta )(3\alpha +\beta )}{(2\alpha +\beta )(\alpha +3\beta )-(\alpha +2\beta )(3\alpha +\beta )}$
$\Rightarrow \frac{1-pq}{p-q}=\frac{-5({\alpha }^2-{\beta }^2)}{-({\alpha }^2-{\beta }^2)}=5$
Ans: D