If log_2 (x) + log_2 (x - 2) = 3, what is the value of x?
Show the answer and explanation
Correct answer
A. 4
Principle or equation
Use the logarithm product rule: log_b M + log_b N = log_b (MN). Then convert to exponential form.
Why this answer is correct
Combine logs: log_2 [x(x - 2)] = 3. Convert: x(x - 2) = 2^3 = 8. Solve x^2 - 2x - 8 = 0 => (x - 4)(x + 2) = 0 => x = 4 or x = -2. Since log arguments must be positive, x = 4 (x = -2 gives log_2(-2) undefined).
Example
Solve log_3 (x) + log_3 (x - 1) = 1. Combine: log_3 [x(x-1)] = 1 => x(x-1) = 3 => x^2 - x - 3 = 0 => x = (1 + sqrt(13))/2 (positive).
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