If `5^x=(0. 5)^y=1000 , then 1/x-1/y=`

Options

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

Explanation

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

### Step 1: Express \( x \) and \( y \) in terms of logarithms
From the equation \( 5^x = 1000 \), we can take logarithms on both sides:
\[ x \log(5) = \log(1000) \]
Thus, we can express \( x \) as:
\[ x = \frac{\log(1000)}{\log(5)} \]

Similarly, from \( (0.5)^y = 1000 \), we take logarithms:
\[ y \log(0.5) = \log(1000) \]
So, we can express \( y \) as:
\[ y = \frac{\log(1000)}{\log(0.5)} \]

### Step 2: Find \( \frac{1}{x} \) 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 3: Calculate \( \frac{1}{x} - \frac{1}{y} \)
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:
\[ \frac{1}{x} - \frac{1}{y} = \frac{\log(5) - \log(0.5)}{\log(1000)} \]

### Step 4: Simplify the expression
Using the property of logarithms \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \):
\[ \log(5) - \log(0.5) = \log\left(\frac{5}{0.5}\right) = \log(10) \]
Thus, we have:
\[ \frac{1}{x} - \frac{1}{y} = \frac{\log(10)}{\log(1000)} \]

### Step 5: Calculate \( \log(1000) \)
Since \( 1000 = 10^3 \), we have:
\[ \log(1000) = 3 \]
Therefore:
\[ \frac{1}{x} - \frac{1}{y} = \frac{\log(10)}{3} \]
And since \( \log(10) = 1 \):
\[ \frac{1}{x} - \frac{1}{y} = \frac{1}{3} \]

### Final Answer
Thus, the final result is:
\[ \frac{1}{x} - \frac{1}{y} = \frac{1}{3} \]