Question

Suppose, $f( x , n )=\displaystyle\sum_{ k =1}^{ n } \log _{ x }\left(\frac{ k }{ x }\right)$, then the value of $x$ satisfying the equation $f( x , 10)=f( x , 11)$, is

• A. 9
• B. 10
• C. 11
• D. none

Answer 1

Answer: (C)

$f ( x , n )=\displaystyle\sum_{ k =1}^{ n } \log _{ x }\left(\frac{ k }{ x }\right)=\log _{ x }\left(\frac{1}{ x }\right)+\log _{ x }\left(\frac{2}{ x }\right)+\ldots \ldots \log _{ x }\left(\frac{ n }{ x }\right)=\log _{ x }\left(\frac{ n !}{ x ^{ n }}\right) $
$\text { given: } f ( x , 10)= f ( x , 11) \Rightarrow \log _{ x }\left(\frac{10 !}{ x ^{10}}\right)=\log _{ x }\left(\frac{11 !}{ x ^{11}}\right) \Rightarrow \frac{10 !}{ x ^{10}}=\frac{11 !}{ x ^{11}} \Rightarrow x =11$