Question

If $lo{g}_{|sinx|}(x^2-8x+23)>\frac{3}{lo{g}_2|sinx|}$

• A. $x\in (3,\pi )\cup (\pi ,\frac{3\pi }{2})\cup (\frac{3\pi }{2},5)$
• B. $x\in (3,\pi )\cup (\pi ,5)$
• C. $x\in (3,\frac{5\pi }{2})$
• D. None of these.

Answer 1

The correct option is A

$x\in (3,\pi )\cup (\pi ,\frac{3\pi }{2})\cup (\frac{3\pi }{2},5)$

The given inequality can be written as $\frac{lo{g}_2(x^2-8x+23)}{lo{g}_2|sinx|}>\frac{3}{lo{g}_2|sinx|}$

As $|sinx|$ can only take values $0<|sinx|<1$ the value of
$lo{g}_2|sinx|$ is negative.

$⟹lo{g}_2(x^2-8x+23)<3\Rightarrow x^2-8x+23<2^3=8\Rightarrow x^2-8x+15<0\Rightarrow (x-5)(x-3)<0\Rightarrow 3<x<5$

But the terms in the inequality are meaningful if $|sinx|\ne 0,1$
So, $\text{/}\in \frac{n\pi }{2}$.
Hence $x\in (3,\pi )\cup (\pi ,\frac{3\pi }{2})\cup (\frac{3\pi }{2},5).$