Let `leftrightarrow` be an operation on p and q defined as p `leftrightarrow` q = `(2^(p)-q)`. If p `leftrightarrow` 12 = p, then find the value of p.

Options

A. 2
B. 4
C. 8
D. 12

Answer

To solve the problem step by step, we will start by analyzing the operation defined in the question and then solve for the value of \( p \).

### Step 1: Understand the operation
The operation \( p \leftrightarrow q \) is defined as:
\[ p \leftrightarrow q = 2^p - q \]
We are given that \( p \leftrightarrow 12 = p \).

### Step 2: Set up the equation
From the definition of the operation, we can substitute \( q = 12 \):
\[ p \leftrightarrow 12 = 2^p - 12 \]
According to the problem, this is equal to \( p \):
\[ 2^p - 12 = p \]

### Step 3: Rearrange the equation
We can rearrange the equation to isolate the exponential term:
\[ 2^p = p + 12 \]

### Step 4: Solve for \( p \)
Now, we need to find the value of \( p \) that satisfies this equation. Since it involves an exponential function, we can test integer values of \( p \).

1. **Test \( p = 2 \)**:
\[ 2^2 = 4 \quad \text{and} \quad 2 + 12 = 14 \quad \Rightarrow \quad 4 \neq 14 \]
(Not a solution)

2. **Test \( p = 4 \)**:
\[ 2^4 = 16 \quad \text{and} \quad 4 + 12 = 16 \quad \Rightarrow \quad 16 = 16 \]
(This is a solution)

3. **Test \( p = 8 \)**:
\[ 2^8 = 256 \quad \text{and} \quad 8 + 12 = 20 \quad \Rightarrow \quad 256 \neq 20 \]
(Not a solution)

4. **Test \( p = 12 \)**:
\[ 2^{12} = 4096 \quad \text{and} \quad 12 + 12 = 24 \quad \Rightarrow \quad 4096 \neq 24 \]
(Not a solution)

### Conclusion
The only value of \( p \) that satisfies the equation \( 2^p = p + 12 \) is:
\[ \boxed{4} \]