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 \( 5^x = (0.5)^y = 1000 \) and find the value of \( \left( \frac{1}{x} - \frac{1}{y} \right) \), we can follow these steps:

### Step 1: Find \( x \) from \( 5^x = 1000 \)

We start with the equation:
\[ 5^x = 1000 \]
Taking logarithm on both sides:
\[ \log(5^x) = \log(1000) \]
Using the logarithmic identity \( \log(a^b) = b \cdot \log(a) \):
\[ x \cdot \log(5) = \log(1000) \]
Now, we can express \( x \):
\[ x = \frac{\log(1000)}{\log(5)} \]

### Step 2: Find \( y \) from \( (0.5)^y = 1000 \)

Next, we consider the second part of the equation:
\[ (0.5)^y = 1000 \]
Taking logarithm on both sides:
\[ \log((0.5)^y) = \log(1000) \]
Again using the logarithmic identity:
\[ y \cdot \log(0.5) = \log(1000) \]
Now, we can express \( y \):
\[ y = \frac{\log(1000)}{\log(0.5)} \]

### Step 3: Calculate \( \frac{1}{x} \) and \( \frac{1}{y} \)

Now, we need to find \( \frac{1}{x} \) and \( \frac{1}{y} \):
\[ \frac{1}{x} = \frac{\log(5)}{\log(1000)} \]
\[ \frac{1}{y} = \frac{\log(0.5)}{\log(1000)} \]

### Step 4: Find \( \frac{1}{x} - \frac{1}{y} \)

Now, we can find \( \frac{1}{x} - \frac{1}{y} \):
\[ \frac{1}{x} - \frac{1}{y} = \frac{\log(5)}{\log(1000)} - \frac{\log(0.5)}{\log(1000)} \]
Combining the fractions:
\[ = \frac{\log(5) - \log(0.5)}{\log(1000)} \]
Using the property of logarithms \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \):
\[ = \frac{\log\left(\frac{5}{0.5}\right)}{\log(1000)} \]
Calculating \( \frac{5}{0.5} \):
\[ \frac{5}{0.5} = 10 \]
Thus:
\[ = \frac{\log(10)}{\log(1000)} \]

### Step 5: Simplify \( \log(1000) \)

We know that \( 1000 = 10^3 \), so:
\[ \log(1000) = 3 \cdot \log(10) = 3 \]
Therefore:
\[ \frac{1}{x} - \frac{1}{y} = \frac{\log(10)}{3} \]
Since \( \log(10) = 1 \):
\[ = \frac{1}{3} \]

### Final Answer

Thus, the value of \( \left( \frac{1}{x} - \frac{1}{y} \right) \) is:
\[ \frac{1}{3} \]