The value of `N=((log)_5 250)/((log)_(50)5)-((log)_5 10)/((log)_(1250)5)` is...........

Explanation

To find the value of \( N = \frac{\log_5 250}{\log_{50} 5} - \frac{\log_5 10}{\log_{1250} 5} \), we can use the properties of logarithms to simplify the expression step by step.

### Step 1: Rewrite the logarithms using the change of base formula
Using the change of base formula, we know that:
\[ \log_a b = \frac{1}{\log_b a} \]
Thus, we can rewrite \( \log_{50} 5 \) and \( \log_{1250} 5 \):
\[ \log_{50} 5 = \frac{1}{\log_5 50} \quad \text{and} \quad \log_{1250} 5 = \frac{1}{\log_5 1250} \]
So, we can rewrite \( N \) as:
\[ N = \log_5 250 \cdot \log_5 50 - \log_5 10 \cdot \log_5 1250 \]

### Step 2: Simplify \( \log_5 250 \) and \( \log_5 10 \)
Next, we simplify \( \log_5 250 \) and \( \log_5 10 \):
\[ \log_5 250 = \log_5 (5^3 \cdot 2) = \log_5 5^3 + \log_5 2 = 3 + \log_5 2 \]
\[ \log_5 10 = \log_5 (5 \cdot 2) = \log_5 5 + \log_5 2 = 1 + \log_5 2 \]

### Step 3: Simplify \( \log_5 50 \) and \( \log_5 1250 \)
Now we simplify \( \log_5 50 \) and \( \log_5 1250 \):
\[ \log_5 50 = \log_5 (5^2 \cdot 2) = \log_5 5^2 + \log_5 2 = 2 + \log_5 2 \]
\[ \log_5 1250 = \log_5 (5^4 \cdot 2) = \log_5 5^4 + \log_5 2 = 4 + \log_5 2 \]

### Step 4: Substitute back into \( N \)
Now substituting these values back into \( N \):
\[ N = (3 + \log_5 2)(2 + \log_5 2) - (1 + \log_5 2)(4 + \log_5 2) \]

### Step 5: Expand both products
Expanding both products:
\[ N = (3 \cdot 2 + 3 \log_5 2 + 2 \log_5 2 + (\log_5 2)^2) - (1 \cdot 4 + 1 \log_5 2 + 4 \log_5 2 + (\log_5 2)^2) \]
\[ N = (6 + 5 \log_5 2 + (\log_5 2)^2) - (4 + 5 \log_5 2 + (\log_5 2)^2) \]

### Step 6: Simplify the expression
Now, simplifying the expression:
\[ N = 6 - 4 + 5 \log_5 2 - 5 \log_5 2 + (\log_5 2)^2 - (\log_5 2)^2 \]
\[ N = 2 \]

### Final Answer
Thus, the value of \( N \) is:
\[ \boxed{2} \]