Correct Answer - Option 3 : Empty set

**General points: **

- sin(x) is the function having a range between -1 and +1.
- Log(x) is defined only when x is positive and greater than zero.

log (sin(x)) is defined only when 0 < sin(x) ≤ 1, and then range will be (−∞,0]

So, log [log (sin(x))] is undefined as the logarithm of non-positive numbers isn’t defined for real numbers.

Hence, Domain of log [ log (sin(x))]: ∅ (empty set)

Also, Range of log [log (sin(x))]: ∅ (empty set)