If `(21. 4)^a=(0. 00214)^b=100` , then the value of `1/a-1/b` is 0 (b) 1 (c) 2 (d) 4

Options

A. 0
B. 1
C. 2
D. 4

Explanation

To solve the equation \( (21.4)^a = (0.00214)^b = 100 \) and find the value of \( \frac{1}{a} - \frac{1}{b} \), we can follow these steps:

### Step 1: Take logarithm of the first equation
Starting with the first part of the equation:
\[ (21.4)^a = 100 \]
Taking logarithm on both sides:
\[ \log((21.4)^a) = \log(100) \]

### Step 2: Apply the power rule of logarithms
Using the property of logarithms \( \log(m^n) = n \log(m) \):
\[ a \log(21.4) = \log(100) \]

### Step 3: Substitute the value of \( \log(100) \)
Since \( \log(100) = 2 \):
\[ a \log(21.4) = 2 \]
From this, we can express \( \log(21.4) \):
\[ \log(21.4) = \frac{2}{a} \tag{1} \]

### Step 4: Take logarithm of the second equation
Now, consider the second part of the equation:
\[ (0.00214)^b = 100 \]
Taking logarithm on both sides:
\[ \log((0.00214)^b) = \log(100) \]

### Step 5: Apply the power rule of logarithms again
Using the property of logarithms:
\[ b \log(0.00214) = \log(100) \]
Substituting \( \log(100) = 2 \):
\[ b \log(0.00214) = 2 \]
From this, we can express \( \log(0.00214) \):
\[ \log(0.00214) = \frac{2}{b} \tag{2} \]

### Step 6: Relate \( \log(0.00214) \) to \( \log(21.4) \)
We can express \( 0.00214 \) as:
\[ 0.00214 = \frac{21.4}{10000} \]
Thus:
\[ \log(0.00214) = \log\left(\frac{21.4}{10000}\right) = \log(21.4) - \log(10000) \]
Since \( \log(10000) = 4 \):
\[ \log(0.00214) = \log(21.4) - 4 \]

### Step 7: Substitute \( \log(21.4) \) from (1)
Substituting \( \log(21.4) \) from equation (1):
\[ \log(0.00214) = \frac{2}{a} - 4 \]

### Step 8: Set the two expressions for \( \log(0.00214) \) equal
Now we have two expressions for \( \log(0.00214) \):
\[ \frac{2}{b} = \frac{2}{a} - 4 \]

### Step 9: Solve for \( \frac{1}{a} - \frac{1}{b} \)
Rearranging gives:
\[ \frac{2}{b} - \frac{2}{a} = -4 \]
Multiplying through by \( ab \):
\[ 2a - 2b = -4ab \]
Rearranging gives:
\[ 2a + 4ab - 2b = 0 \]
Factoring out gives:
\[ 2(a + 2ab - b) = 0 \]
Thus:
\[ a + 2ab - b = 0 \]
Rearranging gives:
\[ \frac{1}{a} - \frac{1}{b} = 2 \]

### Final Answer
Thus, the value of \( \frac{1}{a} - \frac{1}{b} \) is \( 2 \).