Question:
The Range of the function \(y=\log (\sin x)\), defined for all \(x\) such that \(\sin x>0\) is ...
The Range of the function \(y=\log (\sin x)\), defined for all \(x\) such that \(\sin x>0\) is ...
Show Hint
Since sin x lies in (0, 1], take the logarithm of this interval.
Updated On: Nov 12, 2026
- \((-\infty ,0]\)
- \((-\infty ,0)\)
- \(R\)
- \((0,\infty )\)
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept
The function is defined only where \(\sin x>0\).
Also \(\sin x\le1\).
Step 2: Range of the Inner Function
So \(0<\sin x\le1\).
Step 3: Take the Logarithm
The logarithm (base greater than 1) is increasing.
As \(\sin x\to0^{+}\), \(\log(\sin x)\to-\infty\).
At \(\sin x=1\), \(\log 1=0\).
Step 4: Compare the options
So \(y\in(-\infty,0]\), option (A).
Final Answer:
The range is all values up to and including zero. \[ \boxed{\text{(A) }(-\infty ,0]} \]
The function is defined only where \(\sin x>0\).
Also \(\sin x\le1\).
Step 2: Range of the Inner Function
So \(0<\sin x\le1\).
Step 3: Take the Logarithm
The logarithm (base greater than 1) is increasing.
As \(\sin x\to0^{+}\), \(\log(\sin x)\to-\infty\).
At \(\sin x=1\), \(\log 1=0\).
Step 4: Compare the options
So \(y\in(-\infty,0]\), option (A).
Final Answer:
The range is all values up to and including zero. \[ \boxed{\text{(A) }(-\infty ,0]} \]
Was this answer helpful?
0
0
Top MHT CET Mathematics Questions
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- If $\int\frac{f\left(x\right)}{log \left(sin\,x\right)}dx = log\left[log\,sin\,x\right]+c$ then $f\left(x\right)=$
- If $\int\limits^{K}_0 \frac{dx}{2 + 18 x^2} = \frac{\pi}{24}$, then the value of K is
- If $\int^{\pi/2}_{0} \log\cos x dx =\frac{\pi}{2} \log\left(\frac{1}{2}\right)$ then $ \int^{\pi/2}_{0} \log\sec x dx = $
- The point on the curve $y = \sqrt{x - 1}$ where the tangent is perpendicular to the line $2x + y - 5 = 0 $ is
View More Questions
Top MHT CET Functions Questions
- If $f : R - \{2\} \to R$ is a function defined by $f(x) = \frac{x^2 - 4}{x - 2}$ , then its range is
- Find the function $ f(x_1, x_2, x_3) $ satisfying $ f(x_1, x_2, x_3) = 1 $ at $ x_1 = 1, x_2 = x_3 = 0 $ .
- If \( \mathbf{a} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \) and \( \mathbf{b} = 2\mathbf{i} - 3\mathbf{j} - 5\mathbf{k} \), then:
- If \( f(x) = |x| - |1| \), then points where \( f(x) \) is not differentiable, is/are:
- The point on the line \( 4x - y - 2 = 0 \) which is equidistant from the points \( (-5, 6) \) and \( (3, 2) \) is
View More Questions
Top MHT CET Questions
- During r- DNA technology, which one of the following enzymes is used for cleaving DNA molecule ?
- In non uniform circular motion, the ratio of tangential to radial acceleration is (r = radius of circle, $v =$ speed of the particle, $\alpha =$ angular acceleration)
- The temperature of $32^{\circ}C$ is equivalent to
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- The heat of formation of water is $ 260\, kJ $ . How much $ H_2O $ is decomposed by $ 130\, kJ $ of heat ?
View More Questions