Question

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

Answer 1

As, $ 175=5^2 \times 7 ; 245=5 \times 7^2 ; 875=5^3 \times 7 ; 1715=5 \times 7^3$
Now, $\log _{\left(5 \times 7^2\right)}\left(5^2 \times 7\right)= a \Rightarrow \frac{2+\log _5 7}{1+2 \log _5 7}= a$ ...(1)
Also, $ \log _{\left(5 \times 7^3\right)}\left(5^3 \times 7\right)= a \Rightarrow \frac{3+\log _5 7}{1+3 \log _5 7}= b$ ....(2)
$\therefore $ from(1), we get
$2+\log _5 7=a+2 a \cdot \log _5 7 \Rightarrow(2-a)=(2 a-1) \log _5 7 \Rightarrow \log _5 7=\left(\frac{2-a}{2 a-1}\right)$ ....(3)
Also, from (2), we get
$3+\log _5 7= b +3 b \log _5 7 \Rightarrow(3- b )=(3 b -1) \log _5 7 \Rightarrow \log _5 7=\left(\frac{3-b}{3 b-1}\right)$ ....(4)
$\therefore $ From (3) and (4), we get
$ \frac{2-a}{2 a-1}=\frac{3-b}{3 b-1} \Rightarrow(2-a)(3 b-1)=(3-b)(2 a-1) $
$\Rightarrow 6 b-2-3 a b+a=6 a-3-2 a b+b \Rightarrow 1=5 a-5 b+a b \Rightarrow\left(\frac{1-a b}{(a-b)}\right)=5$