`x^((log)_5x)>5` implies `x in (0,oo)` (b) (0,1/5)U(5,∞) (c) (2,2.5) (d) (0,2.5)

Options

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

Explanation

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

### Step-by-Step Solution:

1. **Take Logarithm on Both Sides**:
We start by taking the logarithm (base 5) of both sides of the inequality:
\[ \log_5 (x^{\log_5 x}) > \log_5 5 \]

2. **Simplify the Right Side**:
Since \( \log_5 5 = 1 \), we can rewrite the inequality as:
\[ \log_5 x \cdot \log_5 x > 1 \]
This simplifies to:
\[ (\log_5 x)^2 > 1 \]

3. **Rearranging the Inequality**:
We can express this as:
\[ \log_5 x > 1 \quad \text{or} \quad \log_5 x < -1 \]

4. **Solving the First Inequality**:
For \( \log_5 x > 1 \):
\[ x > 5^1 \implies x > 5 \]

5. **Solving the Second Inequality**:
For \( \log_5 x < -1 \):
\[ x < 5^{-1} \implies x < \frac{1}{5} \]

6. **Combining the Solutions**:
We have two intervals from our inequalities:
- \( x < \frac{1}{5} \)
- \( x > 5 \)

Additionally, since \( x \) must be positive for the logarithm to be defined, we combine these intervals:
\[ x \in (0, \frac{1}{5}) \cup (5, \infty) \]

### Final Solution:
Thus, the solution to the inequality \( x^{\log_5 x} > 5 \) is:
\[ x \in (0, \frac{1}{5}) \cup (5, \infty) \]