If log_2(x) + log_2(x - 3) = 2, find the valid real solution for x.
Answer and explanation
Correct answer: A. 4
- Step 1: Apply the product rule for logarithms: log_2(x(x - 3)) = 2.
- Step 2: Convert to exponential form: x(x - 3) = 2^2 = 4 => x^2 - 3x - 4 = 0.
- Step 3: Factor the quadratic: (x - 4)(x + 1) = 0, giving x = 4 or x = -1.
- Step 4: Check domain: logarithms require strictly positive arguments. x = -1 is extraneous. Thus, x = 4.
Answer
4.
Why not the others: Option 1 yields negative arguments (1 - 3 = -2). Option -1 is an extraneous algebraic root rejected by the logarithm domain. Option 2 gives log_2(2) + log_2(-1), which is undefined.
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