If `a^b =b^a` and a = 2b then value of b is

Options

A. a)1
B. b)`-1`
C. c)2
D. d)`-2`

Answer

To solve the equation \( a^b = b^a \) given that \( a = 2b \), we can follow these steps:

### Step 1: Substitute \( a \) in the equation
Given that \( a = 2b \), we can substitute \( a \) in the equation \( a^b = b^a \):
\[ (2b)^b = b^{2b} \]

### Step 2: Rewrite the equation
Now, we can rewrite the equation as:
\[ 2^b \cdot b^b = b^{2b} \]

### Step 3: Simplify the equation
Next, we can simplify the equation by dividing both sides by \( b^b \) (assuming \( b \neq 0 \)):
\[ 2^b = b^b \]

### Step 4: Take logarithm on both sides
To solve for \( b \), we can take the logarithm of both sides:
\[ \log(2^b) = \log(b^b) \]

### Step 5: Apply the power rule of logarithms
Using the power rule of logarithms, we can simplify this to:
\[ b \log(2) = b \log(b) \]

### Step 6: Divide by \( b \) (assuming \( b \neq 0 \))
Assuming \( b \neq 0 \), we can divide both sides by \( b \):
\[ \log(2) = \log(b) \]

### Step 7: Exponentiate both sides
Exponentiating both sides gives us:
\[ b = 2 \]

### Conclusion
Thus, the value of \( b \) is:
\[ \boxed{2} \]

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