If `(4.2)^(x) = (0.42)^(y) = 100, " then "(1)/(x) -(1)/(y)=`

Options

A. 1
B. 2
C. `(1)/(2)`
D. `-1`

Answer

To solve the equation \( (4.2)^x = (0.42)^y = 100 \) and find \( \frac{1}{x} - \frac{1}{y} \), we can follow these steps:

### Step 1: Set up the equations
From the given equation, we can write:
1. \( (4.2)^x = 100 \)
2. \( (0.42)^y = 100 \)

### Step 2: Take logarithms
Taking logarithm (base 10) on both sides of the first equation:
\[ \log(4.2^x) = \log(100) \]
Using the property of logarithms, we can rewrite this as:
\[ x \cdot \log(4.2) = \log(100) \]
Since \( \log(100) = 2 \) (because \( 100 = 10^2 \)), we have:
\[ x \cdot \log(4.2) = 2 \]
Thus, we can express \( x \) as:
\[ x = \frac{2}{\log(4.2)} \]

### Step 3: Repeat for the second equation
Now, taking logarithm (base 10) on both sides of the second equation:
\[ \log(0.42^y) = \log(100) \]
This can be rewritten as:
\[ y \cdot \log(0.42) = \log(100) \]
Again, since \( \log(100) = 2 \), we have:
\[ y \cdot \log(0.42) = 2 \]
Thus, we can express \( y \) as:
\[ y = \frac{2}{\log(0.42)} \]

### Step 4: Find \( \frac{1}{x} - \frac{1}{y} \)
Now, we need to find \( \frac{1}{x} - \frac{1}{y} \):
\[ \frac{1}{x} = \frac{\log(4.2)}{2} \]
\[ \frac{1}{y} = \frac{\log(0.42)}{2} \]
Thus,
\[ \frac{1}{x} - \frac{1}{y} = \frac{\log(4.2)}{2} - \frac{\log(0.42)}{2} \]
This can be simplified as:
\[ \frac{1}{x} - \frac{1}{y} = \frac{1}{2} \left( \log(4.2) - \log(0.42) \right) \]

### Step 5: Use the property of logarithms
Using the property of logarithms that states \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \), we have:
\[ \log(4.2) - \log(0.42) = \log\left(\frac{4.2}{0.42}\right) \]
Calculating \( \frac{4.2}{0.42} \):
\[ \frac{4.2}{0.42} = 10 \]
Thus,
\[ \log(4.2) - \log(0.42) = \log(10) = 1 \]

### Step 6: Final calculation
Substituting back, we find:
\[ \frac{1}{x} - \frac{1}{y} = \frac{1}{2} \cdot 1 = \frac{1}{2} \]

### Conclusion
The final answer is:
\[ \frac{1}{x} - \frac{1}{y} = \frac{1}{2} \]