A) 2
B) 3
C) 5
D) 7
Correct Answer: C
Solution :
We have, \[{{\log }_{245}}175=a\] | |
\[\Rightarrow \] \[\frac{\log 175}{\log 245}=a\] | |
\[\Rightarrow \] \[\frac{2\log 5+\log 7}{\log 5+2\log 7}=a\] | |
\[\Rightarrow \] \[\frac{2\log 5+\log 7+\log 5+2\log 7}{2\log 5+2\log 7-\log 5-2\log 7}=\frac{a+1}{a-1}\] | |
\[\Rightarrow \] \[3\left( \frac{\log 5+\log 7}{\log 5-\log 7} \right)=\frac{a+1}{a-1}\] | ? (i) |
and \[{{\log }_{175}}875=b\] |
\[\Rightarrow \] \[\frac{\log 875}{\log 1715}=b\]\[\Rightarrow \]\[\frac{3\log 5+\log 7}{\log 5+3\log 7}=b\] |
Apply componendo and dividendo, we get \[2\left( \frac{\log 5+\log 7}{\log 5-\log 7} \right)=\frac{b+1}{b-1}\] |
From Eqs. (i) and (ii), we get \[\frac{1}{3}\left( \frac{a+1}{a-1} \right)=\frac{1}{2}\left( \frac{b+1}{b-1} \right)\] |
\[\Rightarrow \] \[2\,(a+1)(b-1)=3\,(b+1)(a-1)\] |
\[\Rightarrow \] \[2ab+2b-2a-2=3ab+3a-3b-3\] |
\[\Rightarrow \] \[1-ab=5\,(a-b)\] |
\[\Rightarrow \] \[\frac{1-ab}{a-b}=5\] |
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