Question

Let $f\left(x\right)=\log \left({\log }_{\frac{1}{3}}\right({\log }_7(\sin x+a)\left)\right)$ be defined for every real value of $x$, then the possible value of $a$ is

• A. $7$
• B. $8$
• C. $9$
• D. $6$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $7$
B $8$
C $9$

$f\left(x\right)=\log \left({\log }_{\frac{1}{3}}\right({\log }_7(\sin x+a)\left)\right)$
${\log }_{\frac{1}{3}}\left({\log }_7\right(\sin x+a\left)\right)>0$
$\left({\log }_7\right(\sin x+a\left)\right)>1$
$\sin x+a>7$
$\sin x\in [-1,1]$
$\Rightarrow 7<\sin x+a<1+a$

$\Rightarrow 7<1+a\Rightarrow a>6$
$a=7,8,9$