If `a^b = b^a, (a != b) > 1`, then the value of `(a div b)` is :

Options

A. 5
B. 6
C. 7
D. does not exist

Answer

To solve the equation \( a^b = b^a \) with the condition that \( a \neq b \) and both \( a \) and \( b \) are greater than 1, we can follow these steps:

### Step 1: Take the logarithm of both sides
We start by taking the logarithm of both sides of the equation:

\[ \log(a^b) = \log(b^a) \]

### Step 2: Apply the power rule of logarithms
Using the power rule of logarithms, we can rewrite the equation as:

\[ b \cdot \log(a) = a \cdot \log(b) \]

### Step 3: Rearrange the equation
Now, we can rearrange this equation to isolate the ratio of \( a \) and \( b \):

\[ \frac{\log(a)}{a} = \frac{\log(b)}{b} \]

### Step 4: Let \( k \) be the common value
Let \( k \) be the common value of \( \frac{\log(a)}{a} \) and \( \frac{\log(b)}{b} \):

\[ \log(a) = k \cdot a \quad \text{and} \quad \log(b) = k \cdot b \]

### Step 5: Divide the equations
Now, we can divide the two equations:

\[ \frac{\log(a)}{\log(b)} = \frac{k \cdot a}{k \cdot b} = \frac{a}{b} \]

### Step 6: Solve for \( \frac{a}{b} \)
This gives us:

\[ \frac{a}{b} = \frac{\log(a)}{\log(b)} \]

### Step 7: Identify specific values
To find specific values of \( a \) and \( b \), we can use known pairs that satisfy \( a^b = b^a \). A well-known pair is \( (2, 4) \) or \( (4, 2) \).

### Step 8: Calculate \( \frac{a}{b} \)
For \( a = 4 \) and \( b = 2 \):

\[ \frac{a}{b} = \frac{4}{2} = 2 \]

### Conclusion
Thus, the value of \( \frac{a}{b} \) is \( 2 \).

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