Question

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

Answer 1

$a=\frac{{\log }_{5}175}{{\log }_{5}245}=\frac{{\log }_{5}25+{\log }_{5}7}{{\log }_{5}5+{\log }_{5}49}\Rightarrow a=\frac{2+{\log }_{5}7}{1+2{\log }_{5}7}$
$\Rightarrow a+2a{\log }_{5}7=2+{\log }_{5}7$
$\Rightarrow {\log }_{5}7=\frac{a-2}{1-2a}\cdots \left(1\right)$

$b=\frac{{\log }_{5}875}{{\log }_{5}1715}=\frac{3+{\log }_{5}7}{1+3{\log }_{5}7}$
$\Rightarrow b+3b{\log }_{5}7=3+{\log }_{5}7$
$\Rightarrow {\log }_{5}7=\frac{b-3}{1-3b}\cdots \left(2\right)$

From equation $\left(1\right)$ and $\left(2\right)$,
$\frac{a-2}{1-2a}=\frac{b-3}{1-3b}\Rightarrow a-3ab-2+6b=b-3-2ab+6a\Rightarrow 1-ab=5(a-b)$
$\Rightarrow \frac{1-ab}{a-b}=5$