If `f(x)=2^(3x-5)` , find `f^(-1)(16)`

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To find \( f^{-1}(16) \) for the function \( f(x) = 2^{3x - 5} \), we need to determine the value of \( x \) such that \( f(x) = 16 \).

### Step-by-Step Solution:

1. **Set the function equal to 16**:
\[ f(x) = 16 \implies 2^{3x - 5} = 16 \]

2. **Express 16 as a power of 2**:
\[ 16 = 2^4 \]
So we can rewrite the equation as:
\[ 2^{3x - 5} = 2^4 \]

3. **Since the bases are the same, equate the exponents**:
\[ 3x - 5 = 4 \]

4. **Solve for \( x \)**:
- Add 5 to both sides:
\[ 3x = 4 + 5 \]
\[ 3x = 9 \]
- Divide both sides by 3:
\[ x = \frac{9}{3} = 3 \]

5. **Conclusion**:
Therefore, \( f^{-1}(16) = 3 \).