If `p=log_(245)175` and `q=log_(1715) 875`, then `(1-pq)/(p-q)` =

Options

A. `1`
B. `2`
C. `3`
D. `5`

Answer

To solve the problem, we need to find the value of \((1 - pq) / (p - q)\) where \(p = \log_{245} 175\) and \(q = \log_{1715} 875\).

### Step 1: Express \(p\) in terms of logarithms with a common base
Using the change of base formula:
\[ p = \frac{\log_{5} 175}{\log_{5} 245} \]

### Step 2: Factor \(175\) and \(245\)
We can express \(175\) and \(245\) in terms of their prime factors:
\[ 175 = 25 \times 7 = 5^2 \times 7 \]
\[ 245 = 5 \times 49 = 5 \times 7^2 \]

### Step 3: Substitute the factors into the logarithms
Now we can rewrite \(p\):
\[ p = \frac{\log_{5} (5^2 \times 7)}{\log_{5} (5 \times 7^2)} = \frac{\log_{5} 5^2 + \log_{5} 7}{\log_{5} 5 + \log_{5} 7^2} \]
Using the properties of logarithms:
\[ p = \frac{2 + \log_{5} 7}{1 + 2\log_{5} 7} \]

### Step 4: Express \(q\) in terms of logarithms with a common base
Similarly, for \(q\):
\[ q = \frac{\log_{5} 875}{\log_{5} 1715} \]

### Step 5: Factor \(875\) and \(1715\)
We can express \(875\) and \(1715\) in terms of their prime factors:
\[ 875 = 125 \times 7 = 5^3 \times 7 \]
\[ 1715 = 245 \times 7 = 5 \times 7^3 \]

### Step 6: Substitute the factors into the logarithms
Now we can rewrite \(q\):
\[ q = \frac{\log_{5} (5^3 \times 7)}{\log_{5} (5 \times 7^3)} = \frac{\log_{5} 5^3 + \log_{5} 7}{\log_{5} 5 + \log_{5} 7^3} \]
Using the properties of logarithms:
\[ q = \frac{3 + \log_{5} 7}{1 + 3\log_{5} 7} \]

### Step 7: Substitute \(p\) and \(q\) into the expression \((1 - pq) / (p - q)\)
Now we need to compute \(pq\):
\[ pq = \left(\frac{2 + \log_{5} 7}{1 + 2\log_{5} 7}\right) \left(\frac{3 + \log_{5} 7}{1 + 3\log_{5} 7}\right) \]

### Step 8: Calculate \(1 - pq\) and \(p - q\)
We will compute the numerator and denominator separately:
1. **Numerator**: \(1 - pq\)
2. **Denominator**: \(p - q\)

### Step 9: Simplify the expression
After substituting and simplifying, we will find the final value of \((1 - pq) / (p - q)\).

### Final Result
After performing the calculations, we find that:
\[ \frac{1 - pq}{p - q} = 5 \]