If `(31.6)^a=(0.0000316)^b=100` , the value of `1/a-1/b`is

Answer

To solve the problem where \( (31.6)^a = (0.0000316)^b = 100 \), we need to find the value of \( \frac{1}{a} - \frac{1}{b} \). Let's break it down step by step.

### Step 1: Set up the equations
We have two equations based on the given information:
1. \( (31.6)^a = 100 \)
2. \( (0.0000316)^b = 100 \)

### Step 2: Take logarithms
Taking logarithm base 10 of both sides for the first equation:
\[ \log_{10}((31.6)^a) = \log_{10}(100) \]
Using the property of logarithms, we can rewrite this as:
\[ a \cdot \log_{10}(31.6) = \log_{10}(10^2) \]
This simplifies to:
\[ a \cdot \log_{10}(31.6) = 2 \]
Thus, we can express \( a \) as:
\[ a = \frac{2}{\log_{10}(31.6)} \]

### Step 3: Repeat for the second equation
Now, for the second equation, we take logarithm base 10:
\[ \log_{10}((0.0000316)^b) = \log_{10}(100) \]
This can be rewritten as:
\[ b \cdot \log_{10}(0.0000316) = 2 \]
Next, we simplify \( \log_{10}(0.0000316) \):
\[ 0.0000316 = \frac{31.6}{10^5} \implies \log_{10}(0.0000316) = \log_{10}(31.6) - 5 \]
Substituting this back gives:
\[ b \cdot (\log_{10}(31.6) - 5) = 2 \]
Thus, we can express \( b \) as:
\[ b = \frac{2}{\log_{10}(31.6) - 5} \]

### Step 4: Find \( \frac{1}{a} - \frac{1}{b} \)
Now we need to calculate \( \frac{1}{a} - \frac{1}{b} \):
\[ \frac{1}{a} = \frac{\log_{10}(31.6)}{2} \]
\[ \frac{1}{b} = \frac{\log_{10}(31.6) - 5}{2} \]
Now, substituting these into the expression:
\[ \frac{1}{a} - \frac{1}{b} = \frac{\log_{10}(31.6)}{2} - \frac{\log_{10}(31.6) - 5}{2} \]
This simplifies to:
\[ \frac{1}{a} - \frac{1}{b} = \frac{\log_{10}(31.6) - (\log_{10}(31.6) - 5)}{2} = \frac{5}{2} \]

### Final Result
Thus, the value of \( \frac{1}{a} - \frac{1}{b} \) is:
\[ \frac{5}{2} = 2.5 \]