Question

If \( 8^{2x - 4} = 16^{x - 2} \), then \( x = \)

• A. 2
• B. 4
• C. 6
• D. 1

Hint:

When solving exponential equations, express both sides in terms of the same base to simplify the comparison of exponents.

Answer 1

The Correct Option is A

First, express both sides of the equation in terms of powers of 2: \[ 8 = 2^3 \quad \text{and} \quad 16 = 2^4. \] Thus, the equation becomes: \[ (2^3)^{2x - 4} = (2^4)^{x - 2}. \] Simplifying: \[ 2^{3(2x - 4)} = 2^{4(x - 2)} \quad \Rightarrow \quad 2^{6x - 12} = 2^{4x - 8}. \] Since the bases are the same, equate the exponents: \[ 6x - 12 = 4x - 8 \quad \Rightarrow \quad 2x = 4 \quad \Rightarrow \quad x = 2. \]