If log_2 (x) + log_2 (x + 6) = 4, what is the value of x?
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Correct answer
A. 2
Principle or equation
Use the logarithm product rule: log_b (M) + log_b (N) = log_b (MN). Convert to exponential form and solve the resulting quadratic, checking for extraneous solutions.
Why this answer is correct
log_2 [x(x + 6)] = 4 => x(x + 6) = 2^4 = 16 => x^2 + 6x - 16 = 0 => (x + 8)(x - 2) = 0 => x = -8 or x = 2. Since logarithms require x > 0 and x + 6 > 0, x = 2 is the only valid solution.
Example
Solve log_3 (y) + log_3 (y - 2) = 1: y(y - 2) = 3 => y^2 - 2y - 3 = 0 => y = 3 or -1, so y = 3.
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