`x^(log _(5)x) gt 5` implies:

Options

A. `x in (0, infty)`
B. `x in (1, infty)`
C. `x in (1,2)`
D. `x in (0,1//5) cup (5, infty)`

Answer

To solve the inequality \( x^{\log_5 x} > 5 \), we will follow these steps:

### Step 1: Rewrite the inequality using logarithms
We start with the given inequality:
\[ x^{\log_5 x} > 5 \]
Taking the logarithm base 5 of both sides, we get:
\[ \log_5 (x^{\log_5 x}) > \log_5 5 \]
Since \( \log_5 5 = 1 \), the inequality simplifies to:
\[ \log_5 x \cdot \log_5 x > 1 \]
This can be rewritten as:
\[ (\log_5 x)^2 > 1 \]

### Step 2: Solve the quadratic inequality
The inequality \( (\log_5 x)^2 > 1 \) implies:
\[ \log_5 x > 1 \quad \text{or} \quad \log_5 x < -1 \]

### Step 3: Convert back to exponential form
1. For \( \log_5 x > 1 \):
\[ x > 5^1 \implies x > 5 \]
2. For \( \log_5 x < -1 \):
\[ x < 5^{-1} \implies x < \frac{1}{5} \]

### Step 4: Combine the results
The solution to the inequality \( x^{\log_5 x} > 5 \) is:
\[ x < \frac{1}{5} \quad \text{or} \quad x > 5 \]

### Final Answer
Thus, the solution set is:
\[ (0, \frac{1}{5}) \cup (5, \infty) \]