Question

Suppose, $f(x,n)=\sum _{k=1}^n{\log }_x\left(\frac{k}{x}\right)$, 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)

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