Question

The domain of the function $f\left(x\right)=\ln \left\{{\log }_{|\sin x|}\right(x^2-8x+23)-\frac{3}{{\log }_2|\sin x|}\}$ contains which of the following interval(s)?

• A. $(0,3)$
• B. $(\pi ,\frac{3\pi }{2})$
• C. $(\frac{3\pi }{2},5)$
• D. $(5,\infty )$

Answer 1

Answer: (B), (C)

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

Given : $f\left(x\right)=\ln \left\{{\log }_{|\sin x|}\right(x^2-8x+23)-\frac{3}{{\log }_2|\sin x|}\}$

$f\left(x\right)$ is defined if
$\Rightarrow {\log }_{|\sin x|}(x^2-8x+23)-\frac{3}{{\log }_2|\sin x|}>0\Rightarrow {\log }_{|\sin x|}(x^2-8x+23)-{\log }_{|\sin x|}{2}^3>0\Rightarrow {\log }_{|\sin x|}\frac{x^2-8x+23}{8}>0$

Also,
$|\sin x|\ne 0,1$ and $\frac{x^2-8x+23}{8}<1$
$\Rightarrow x^2-8x+15<0\Rightarrow (x-3)(x-5)<0\therefore x\in (3,5)-\{\pi ,\frac{3\pi }{2}\}$

Hence, domain of the function is
$(3,\pi )\cup (\pi ,\frac{3\pi }{2})\cup (\frac{3\pi }{2},5)$