Question

The range of the function \[ y=\log(\sin x) \] where \( \sin x>0 \) is:

• A. \([0,\infty)\)
• B. \((-\infty,0]\)
• C. \((-\infty,\infty)\)
• D. \([-1,1]\)

Hint:

Remember: \[ \log t \le 0 \quad \text{for} \quad 0<t\le1 \]

Answer 1

The Correct Option is B

Concept: For logarithmic functions: \[ \log t \] is defined only when: \[ t>0 \] Also: \[ 0<\sin x\le1 \] for all values where \(\sin x>0\).

Step 1: Find range of \( \sin x \). Given: \[ \sin x>0 \] Hence: \[ 0<\sin x\le1 \]

Step 2: Apply logarithm. Taking logarithm: \[ y=\log(\sin x) \] Now: \[ \log 1=0 \] and as: \[ \sin x\to0^+ \] we get: \[ \log(\sin x)\to -\infty \] Thus: \[ -\infty<y\le0 \]

Step 3: Final range. \[ \boxed{ (-\infty,0] } \]