Which one of following best represents the graph of `y=x^(log_x pi)`

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining which graph best represents the equation \( y = x^{\log_x \pi} \), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the equation:
\[ y = x^{\log_x \pi} \]
Using the logarithmic identity \( a^{\log_b c} = c^{\log_b a} \), we can rewrite the equation as:
\[ y = \pi^{\log_x x} \]

### Step 2: Simplify the Logarithm
Since \( \log_x x = 1 \) (because any logarithm of a number to its own base is 1), we can simplify further:
\[ y = \pi^1 = \pi \]

### Step 3: Determine the Domain
Next, we need to consider the domain of the function. The logarithm \( \log_x \pi \) is defined only when:
1. \( x > 0 \) (the base of the logarithm must be positive)
2. \( x \neq 1 \) (the base cannot be 1)

Thus, the domain of \( x \) is:
\[ x \in (0, 1) \cup (1, \infty) \]

### Step 4: Identify the Range
From our simplification, we see that \( y \) is constant and equal to \( \pi \) for all valid \( x \) in the domain. Therefore, the range of \( y \) is:
\[ y = \pi \]

### Step 5: Graph the Function
The graph of the function \( y = \pi \) is a horizontal line at \( y = \pi \). However, we must note that this line does not include the points where \( x = 1 \) and \( x = 0 \).

### Conclusion
The graph of \( y = x^{\log_x \pi} \) is a horizontal line at \( y = \pi \), extending from \( x = 0 \) to \( x = 1 \) (not including 0 and 1) and from \( x = 1 \) to \( x = \infty \).

### Final Answer
The best representation of the graph of \( y = x^{\log_x \pi} \) is a horizontal line at \( y = \pi \) for \( x \in (0, 1) \cup (1, \infty) \).

---