Solve the exponential equation 4^x - 3 * 2^x - 4 = 0 for real values of x.
Answer and explanation
Correct answer: B. x = 2
- Step 1: Recognize 4^x as (2^x)^2.
- Step 2: Substitute u = 2^x (with u > 0): u^2 - 3u - 4 = 0.
- Step 3: Factor the quadratic: (u - 4)(u + 1) = 0 => u = 4 or u = -1.
- Step 4: Since 2^x must be positive, discard u = -1.
- Step 5: Solve 2^x = 4 = 2^2 => x = 2.
Answer
x = 2.
Why not the others: 'x = 4' confuses u with x; 'x = 1' evaluates 4^1 - 3(2) - 4 = -6 != 0; 'x = -1' attempts to solve 2^x = -1 which has no real solution.
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