Question

Domain of the function \(f(x)=\log_x(\cos x)\), is

• A. \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)-\{1\}\)
• B. \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]-\{1\}\)
• C. \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\)
• D. None of these

Hint:

For \(\log_x(\cos x)\): require \(x>0,\ x\ne 1\) and \(\cos x>0\). Hence domain is union of intervals where \(\cos x>0\) with \(x>0\).

Answer 1

The Correct Option is D

Step 1: Conditions for \(\log_x(\cos x)\) to be defined.
For \(\log_a(b)\) to exist in real numbers: 
\[ a>0,\ a\ne 1,\quad b>0 \] So here: 
\[ x>0,\ x\ne 1,\quad \cos x>0 \] 

Step 2: Solve \(\cos x>0\). 
\[ \cos x>0 \Rightarrow x\in\left(-\frac{\pi}{2}+2k\pi,\frac{\pi}{2}+2k\pi\right) \] But also we need \(x>0\). 
So domain is: 
\[ x\in\left(0,\frac{\pi}{2}\right)\cup\left(\frac{3\pi}{2},\frac{5\pi}{2}\right)\cup\cdots \] and also excluding \(x=1\). 

Step 3: Compare with given options. 
None of the given sets represents this union of intervals. 
Final Answer: \[ \boxed{\text{None of these}} \]