The domain of definition of the function `f(x) = log_(3) {-log_(1//2)(1+(1)/(x^(1//5)))-1}`

Options

A. `(-oo, 1)`
B. `(0, 1 )`
C. `(1, oo)`
D. none of these

Explanation

To find the domain of the function \( f(x) = \log_{3} \left( -\log_{\frac{1}{2}} \left( 1 + \frac{1}{x^{\frac{1}{5}}} \right) - 1 \right) \), we need to ensure that the argument of the logarithm is positive. This means we need to solve the inequality:

\[ -\log_{\frac{1}{2}} \left( 1 + \frac{1}{x^{\frac{1}{5}}} \right) - 1 > 0 \]

### Step 1: Rearranging the Inequality
First, we can rearrange the inequality:

\[ -\log_{\frac{1}{2}} \left( 1 + \frac{1}{x^{\frac{1}{5}}} \right) > 1 \]

### Step 2: Multiplying by -1
Next, we multiply both sides by -1, which reverses the inequality:

\[ \log_{\frac{1}{2}} \left( 1 + \frac{1}{x^{\frac{1}{5}}} \right) < -1 \]

### Step 3: Applying Logarithmic Properties
Using the property of logarithms, we know that if \( \log_{a}(b) < k \), then \( b < a^{k} \). Here, \( a = \frac{1}{2} \) and \( k = -1 \):

\[ 1 + \frac{1}{x^{\frac{1}{5}}} < \left( \frac{1}{2} \right)^{-1} = 2 \]

### Step 4: Isolating the Fraction
Now, we isolate the fraction:

\[ \frac{1}{x^{\frac{1}{5}}} < 2 - 1 \]
\[ \frac{1}{x^{\frac{1}{5}}} < 1 \]

### Step 5: Inverting the Inequality
Taking the reciprocal of both sides (and reversing the inequality):

\[ x^{\frac{1}{5}} > 1 \]

### Step 6: Raising Both Sides to the Power of 5
Now, we raise both sides to the power of 5:

\[ x > 1 \]

### Step 7: Considering the Domain
Since \( x^{\frac{1}{5}} \) is defined for \( x > 0 \), we combine this with our previous result. Therefore, the domain of \( f(x) \) is:

\[ x > 1 \]

### Final Answer
Thus, the domain of the function \( f(x) \) is \( (1, \infty) \).

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