Question

If ${\log }_{245}175=a$, ${\log }_{1715}875=b$, then find value of $\frac{1-ab}{a-b}$.

Answer 1

$a={\log }_{245}175=\frac{\log 175}{\log 245}=\frac{\log \left(5^2.7\right)}{\log \left(7^2.5\right)}=\frac{2\log 5+\log 7}{2\log 7+\log 5}$

$b={\log }_{1715}875=\frac{\log 875}{\log 1715}=\frac{\log \left(5^3.7\right)}{\log \left(7^3.5\right)}=\frac{3\log 5+\log 7}{3\log 7+\log 5}$

Now,

Let $\log 5=p$ and $\log 7=q$

$\Rightarrow a=\frac{2p+q}{2q+p}$

$\Rightarrow b=\frac{3p+q}{3q+p}$

so, $\frac{1-ab}{a-b}$ is given as:

$=\frac{1-\frac{2p+q}{2q+p}\cdot \frac{3p+q}{3q+p}}{\frac{2p+q}{2q+p}-\frac{3p+q}{3q+p}}$

$=\frac{\frac{(2q+p)(3q+p)-(2p+q)(3p+q)}{(2q+p)(3p+q)}}{\frac{(2p+q)(3q+p)-(3p+q)(2q+p)}{(2q+p)(3q+p)}}$

$=\frac{6q^2+3pq+2pq+p^2-6p^2-3pq-2pq-q^2}{6pq+3q^2+2p^2+pq-6pq-3p^2-2q^2-pq}$

$=\frac{5q^2-5p^2}{q^2-p^2}$

$=5$