Question

The Range of the function \(y=\log (\sin x)\), defined for all \(x\) such that \(\sin x>0\) is ...

• A. \((-\infty ,0]\)
• B. \((-\infty ,0)\)
• C. \(R\)
• D. \((0,\infty )\)

Hint:

Since sin x lies in (0, 1], take the logarithm of this interval.

Answer 1

The Correct Option is A

Step 1: Maximum Value
The largest value of $\sin x$ is $1$, so the largest value of $y$ is $\log1=0$, and it is attained.

Step 2: Lower Behaviour
Values of $\sin x$ very close to zero make $y$ arbitrarily negative.

Step 3: Result
The range is $(-\infty,0]$ including 0, option (A) and not (B).

Final Answer:
Zero is attained and values go down without bound, so the range is minus infinity to 0 inclusive. \[ \boxed{\text{(A) }(-\infty ,0]} \]