Solution of inequality `log_(sin x)(log_(3)(log_(0.2)x))lt0` is

Options

A. `[0, 2]`
B. `(0, (1)/(2))`
C. `((1)/(125), (1)/(2))`
D. `(0, (1)/(125))`

Explanation

To solve the inequality \( \log_{\sin x}(\log_{3}(\log_{0.2} x)) < 0 \), we will follow these steps:

### Step 1: Rewrite the logarithm
Using the change of base formula for logarithms, we can rewrite the inequality:
\[ \log_{\sin x}(\log_{3}(\log_{0.2} x)) < 0 \implies \frac{\log(\log_{3}(\log_{0.2} x))}{\log(\sin x)} < 0 \]
This implies that the numerator and denominator must have opposite signs.

### Step 2: Analyze the denominator
The denominator \( \log(\sin x) \) is less than 0 when \( \sin x < 1 \). Since \( \sin x \) is always positive for \( x \in (0, \pi) \), we need to find when \( \sin x = 1 \) which occurs at \( x = \frac{\pi}{2} \). Thus, \( \log(\sin x) < 0 \) for \( x \in (0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi) \).

### Step 3: Analyze the numerator
For the numerator \( \log(\log_{3}(\log_{0.2} x)) < 0 \), we need:
\[ \log_{3}(\log_{0.2} x) < 1 \implies \log_{0.2} x < 3 \]
This can be rewritten using the change of base:
\[ \log_{0.2} x < 3 \implies x < 0.2^3 = 0.008 \]

### Step 4: Combine the results
Now we have two conditions:
1. \( x < 0.008 \)
2. \( x > 0 \)

Thus, the solution to the inequality is:
\[ 0 < x < 0.008 \]

### Final Answer
The solution of the inequality is \( x \in (0, 0.008) \).

---