Given `log_(x)25 - log_(x) 5 = 2 - "log"_(x) (1)/(125)`, find x.

Answer

To solve the equation \( \log_{x} 25 - \log_{x} 5 = 2 - \log_{x} \left( \frac{1}{125} \right) \), we can follow these steps:

### Step 1: Rewrite the logarithms
We start by using the properties of logarithms. We know that:
- \( \log_{b} a - \log_{b} c = \log_{b} \left( \frac{a}{c} \right) \)
- \( \log_{b} (a^n) = n \cdot \log_{b} a \)

Using these properties, we can rewrite the left side:
\[ \log_{x} 25 - \log_{x} 5 = \log_{x} \left( \frac{25}{5} \right) = \log_{x} 5 \]

### Step 2: Simplify the right side
Next, we simplify the right side. We can rewrite \( \frac{1}{125} \) as \( 125^{-1} \) or \( 5^{-3} \):
\[ \log_{x} \left( \frac{1}{125} \right) = \log_{x} (5^{-3}) = -3 \log_{x} 5 \]
Thus, the right side becomes:
\[ 2 - \log_{x} \left( \frac{1}{125} \right) = 2 + 3 \log_{x} 5 \]

### Step 3: Set the two sides equal
Now we can set the simplified left side equal to the simplified right side:
\[ \log_{x} 5 = 2 + 3 \log_{x} 5 \]

### Step 4: Rearrange the equation
To isolate \( \log_{x} 5 \), we can rearrange the equation:
\[ \log_{x} 5 - 3 \log_{x} 5 = 2 \]
This simplifies to:
\[ -2 \log_{x} 5 = 2 \]
Dividing both sides by -2 gives:
\[ \log_{x} 5 = -1 \]

### Step 5: Convert the logarithmic equation to exponential form
Now we convert the logarithmic equation to its exponential form:
\[ x^{-1} = 5 \]
This implies:
\[ x = \frac{1}{5} \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{\frac{1}{5}} \]