`f(x)=log_(x^(2)) 25 and g(x)=log_(x)5.` Then f(x)=g(x) holds for x belonging to

Answer

To solve the problem where \( f(x) = \log_{x^2} 25 \) and \( g(x) = \log_x 5 \), we need to find the values of \( x \) for which \( f(x) = g(x) \).

### Step 1: Rewrite the functions using logarithmic properties
We can use the change of base formula for logarithms. The property states that:
\[ \log_{a^n} b = \frac{1}{n} \log_a b \]
Using this property, we can rewrite \( f(x) \):
\[ f(x) = \log_{x^2} 25 = \frac{1}{2} \log_x 25 \]

### Step 2: Set the equations equal to each other
Now we can set \( f(x) \) equal to \( g(x) \):
\[ \frac{1}{2} \log_x 25 = \log_x 5 \]

### Step 3: Eliminate the fraction
To eliminate the fraction, we can multiply both sides by 2:
\[ \log_x 25 = 2 \log_x 5 \]

### Step 4: Use the property of logarithms
Using the property of logarithms that states \( \log_a b^n = n \log_a b \), we can rewrite the right side:
\[ \log_x 25 = \log_x 5^2 \]

### Step 5: Set the arguments equal to each other
Since the logarithms are equal, we can set their arguments equal to each other:
\[ 25 = 5^2 \]

### Step 6: Solve for x
This equality holds true. Therefore, we can conclude that \( f(x) = g(x) \) for all \( x \) in the domain of the logarithmic functions.

### Step 7: Determine the domain
The domain of \( f(x) \) and \( g(x) \) must be considered:
1. \( x > 0 \) (since the base of the logarithm must be positive)
2. \( x \neq 1 \) (since the base cannot be 1)

Thus, the solution set for \( x \) is:
\[ x \in (0, 1) \cup (1, \infty) \]

### Final Answer
The values of \( x \) for which \( f(x) = g(x) \) holds are:
\[ x \in (0, 1) \cup (1, \infty) \]