If `5^(x)=(0.5)^(y)=1000`, then the value of `(1/x-1/y)` is

Options

A. `1/4`
B. `1/3`
C. `1/2`
D. 1

Answer

To solve the problem, we need to find the value of \( \frac{1}{x} - \frac{1}{y} \) given that \( 5^x = (0.5)^y = 1000 \).

### Step-by-Step Solution:

1. **Express \( 0.5 \) in terms of powers of \( 5 \)**:
\[ 0.5 = \frac{1}{2} = \frac{1}{5^{\log_5{2}}} = 5^{-\log_5{2}} \]
Therefore, we can rewrite the equation as:
\[ 5^x = 5^{-\log_5{2} \cdot y} = 1000 \]

2. **Express \( 1000 \) as a power of \( 5 \)**:
\[ 1000 = 10^3 = (5 \cdot 2)^3 = 5^3 \cdot 2^3 \]
Thus, we can write:
\[ 5^x = 5^3 \cdot 2^3 \]

3. **Set the exponents equal**:
From \( 5^x = 5^3 \cdot 2^3 \), we can equate the exponents:
\[ x = 3 + 3 \cdot \log_5{2} \]

4. **Find \( y \)**:
Since \( (0.5)^y = 1000 \), we can write:
\[ (5^{-\log_5{2}})^y = 1000 \]
This simplifies to:
\[ 5^{-y \cdot \log_5{2}} = 1000 \]
Taking logarithm base \( 5 \):
\[ -y \cdot \log_5{2} = 3 \implies y = -\frac{3}{\log_5{2}} \]

5. **Calculate \( \frac{1}{x} \) and \( \frac{1}{y} \)**:
\[ \frac{1}{x} = \frac{1}{3 + 3 \cdot \log_5{2}} = \frac{1}{3(1 + \log_5{2})} \]
\[ \frac{1}{y} = -\frac{\log_5{2}}{3} \]

6. **Find \( \frac{1}{x} - \frac{1}{y} \)**:
\[ \frac{1}{x} - \frac{1}{y} = \frac{1}{3(1 + \log_5{2})} + \frac{\log_5{2}}{3} \]
To combine these fractions, we find a common denominator:
\[ = \frac{1 + \log_5{2}(1 + \log_5{2})}{3(1 + \log_5{2})} \]

7. **Simplify the expression**:
\[ = \frac{1 + \log_5{2} + (\log_5{2})^2}{3(1 + \log_5{2})} \]

### Final Result:
The value of \( \frac{1}{x} - \frac{1}{y} \) is:
\[ \frac{1 + \log_5{2} + (\log_5{2})^2}{3(1 + \log_5{2})} \]