Question

Which of the following is $NOT$ the solution of the equation $\log (\frac{5}{2}-\frac{1}{x})>\log x$?

• A. $(\frac{1}{2},2)$
• B. $(1,2)$
• C. $(\frac{2}{5},\frac{1}{2})$
• D. $(\frac{1}{2},1)$

Answer 1

Answer: (C)

$(\frac{2}{5},\frac{1}{2})$

$\log (\frac{5}{2}-\frac{1}{x})$ is defined when $x>\frac{2}{5}$....(1)
Similarly, $\log x$ is defined when $x>0$....(2)
From (1) and (2), we get $x>\frac{2}{5}$.....(3)
Now, $\log (\frac{5}{2}-\frac{1}{x})>\log x$
$\Rightarrow \frac{5}{2}-\frac{1}{x}>x$
$\Rightarrow 2x^2-5x+2<0$
$(2x-1)(x-2)<0$
$x\in (\frac{1}{2},2)$
Hence option C is not the solution of the given equation.