If `log_(5)x-log_(5)y = log_(5)4 + log_(5)2 and x - y = 7`, then = ______ .

Options

A. 1
B. 8
C. 7
D. 6

Explanation

To solve the equation \( \log_5 x - \log_5 y = \log_5 4 + \log_5 2 \) and \( x - y = 7 \), we will follow these steps:

### Step 1: Simplify the logarithmic equation
Using the properties of logarithms, we can rewrite the left side:
\[ \log_5 x - \log_5 y = \log_5 \left(\frac{x}{y}\right) \]
For the right side, we can combine the logarithms:
\[ \log_5 4 + \log_5 2 = \log_5 (4 \cdot 2) = \log_5 8 \]
So, we have:
\[ \log_5 \left(\frac{x}{y}\right) = \log_5 8 \]

### Step 2: Remove the logarithm
Since the logarithms are equal, we can set the arguments equal to each other:
\[ \frac{x}{y} = 8 \]
This implies:
\[ x = 8y \]

### Step 3: Substitute into the second equation
We know from the problem statement that \( x - y = 7 \). Now, substituting \( x = 8y \) into this equation gives:
\[ 8y - y = 7 \]
This simplifies to:
\[ 7y = 7 \]

### Step 4: Solve for \( y \)
Dividing both sides by 7:
\[ y = 1 \]

### Step 5: Find \( x \)
Now, substituting \( y = 1 \) back into \( x = 8y \):
\[ x = 8 \cdot 1 = 8 \]

### Conclusion
Thus, the values of \( x \) and \( y \) are:
\[ x = 8 \quad \text{and} \quad y = 1 \]

### Final Answer
The value of \( x \) is \( 8 \).

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