Number of integral values of a for which $$f(x)=log(log13(log7(sinx+a)))$$ be defined for every real value of x is

$$log13(log7(sinx+a))$$>$$0$$ ⇒$$0$$

Functions (XII)

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Question

${Number of integral values of a for which f(x)=log(log}_{\frac{1}{3}} (l o g_{7} (s i n x + a))) b e d e f i n e d f o r e v e r y r e a l v a l u e o f x i s$

Moderate

Solution

${log}_{\frac{1}{3}} (l o g_{7} (s i n x + a)) > 0 \Rightarrow 0 < l o g_{7} (s i n x + a) < 1$
$\Rightarrow 1 < s i n x + a < 7 \Rightarrow 1 - s i n x < a < 7 - s i n x \forall x \in R ∴ 2 < a < 6 a s m i n i m u m o f s i n x = - 1 a n d m a x i m u m o f s i n x = 1$

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