Question

The solution of the inequality ${\log }_x(2x-\frac{3}{4})>2$ is

• A. $\frac{1}{2}<x<\frac{3}{2}$
  
• B. $\frac{3}{2}<x<\frac{1}{2}$
  
• C. $\frac{1}{2}<x<\frac{2}{3}$
  
• D. None of these

Answer 1

The correct option is A $\frac{1}{2}<x<\frac{3}{2}$

${\log }_x(2x-\frac{3}{4})>2$

$\frac{\log (2x-\frac{3}{4})}{\log x}>2$

$\log (2x-\frac{3}{4})>2.\log x$

$\log (2x-\frac{3}{4})>\log x^2$

$2x-\frac{3}{4}>x^2$

$8x-3>4x^2$

$0>4x^2-8x+3$

$0>4x^2-6x-2x+3$

$0>2x(2x-3)-1(2x-3)$

$0>(2x-3)(2x-1)$

$\frac{1}{2}<x<\frac{3}{2}$