The values of x satisfying `x^("log"_(5)) gt5` lie in the interval

Options

A. `(0, oo)`
B. `(0, 1//5) cup (5, oo)`
C. `(1, oo)`
D. (1, 2)

Explanation

To solve the inequality \( x^{\log_5 x} > 5 \), we will analyze the expression step by step.

### Step 1: Rewrite the Inequality
We start with the inequality:
\[ x^{\log_5 x} > 5 \]
We can rewrite \( 5 \) as \( 5^1 \).

### Step 2: Apply Logarithms
Taking logarithm (base 5) on both sides gives us:
\[ \log_5(x^{\log_5 x}) > \log_5(5) \]
Using the property of logarithms, we have:
\[ \log_5 x \cdot \log_5 x > 1 \]
This simplifies to:
\[ (\log_5 x)^2 > 1 \]

### Step 3: Solve the Quadratic Inequality
The inequality \( (\log_5 x)^2 > 1 \) can be solved by considering two cases:
1. \( \log_5 x > 1 \)
2. \( \log_5 x < -1 \)

#### Case 1: \( \log_5 x > 1 \)
This implies:
\[ x > 5^1 \quad \Rightarrow \quad x > 5 \]

#### Case 2: \( \log_5 x < -1 \)
This implies:
\[ x < 5^{-1} \quad \Rightarrow \quad x < \frac{1}{5} \]

### Step 4: Combine the Results
From the two cases, we find:
1. From Case 1: \( x > 5 \)
2. From Case 2: \( x < \frac{1}{5} \)

Thus, the solution set for \( x \) satisfying the inequality \( x^{\log_5 x} > 5 \) is:
\[ x \in \left(0, \frac{1}{5}\right) \cup (5, \infty) \]

### Final Answer
The values of \( x \) satisfying \( x^{\log_5 x} > 5 \) lie in the interval:
\[ \left(0, \frac{1}{5}\right) \cup (5, \infty) \]