ISRO2018-15

Question

The domain of the function $\log (\log \sin(x))$ is:

1. $0<x<$$\pi$
2. $2n$$\pi$$<$$x$$<$$(2n+1)$$\pi$, for $n$ in $N$
3. Empty set
4. None of the above

Answer 1

log( log sin(x) )

-1 <= sinx<= +1

log a is defined for positive values of a,

log sin(x)  is defined for sin(x)= (0,1]

Possible values for  log sin(x) = ($-\infty$ , 0]

Domain of log( log sin(x) )=Not defined

Therefore, Answer (c) Empty Set

Comments

there is a difference between range and domain of a function, Question is asking domain not range.

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* log sin(x)  is defined for sin(x)= (0,1]

Answer 2

Option C is correct.

We can satisfy inner log using $x=\pi / 2$, but can't satisfy outer log at the same time. Thus, empty set.

Answer 3

$\log sin(x)>0 => sin (x)>e^0.$

Now this is impossible because the value of sin cannot be greater than 1.

Therefore, empty set is the answer.