Question

Solve for $x$: ${\log }_2\frac{x-4}{2x+5}<1$

Answer 1

For logarithm to be defined,

$\frac{x-4}{2x+5}>0$

$\Rightarrow \frac{(x-4)(2x+5)}{(2x+5{)}^2}>0$

$\Rightarrow (x-4)(2x+5)>0$

$\Rightarrow x\in (-\infty ,\frac{-5}{2})\cup (4,\infty )$ ...(1)

And $(2x+5)>0$

$\Rightarrow x>\frac{-5}{2}$ ...(2)

Now ${\log }_2\left(\frac{x-4}{2x+5}\right)<1$

$\Rightarrow \frac{x-4}{2x+5}<{\left(2\right)}^1$
$\Rightarrow x-4<2(2x+5)$

$\Rightarrow x-4<4x+10$

$\Rightarrow 3x+14>0$

$\Rightarrow x>\frac{-14}{3}$ ...(3)

From (1), (2) and (3) $x\in (4,\infty )$