If log_2 (x) + log_2 (x + 3) = 2, what is the value of x?
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Correct answer
A. 1
Principle or equation
Use the logarithm product rule: log_b (A) + log_b (B) = log_b (AB). Then convert the logarithmic equation to an exponential equation.
Why this answer is correct
Combine the logs: log_2 [x(x + 3)] = 2. This means x(x + 3) = 2^2 = 4. Solve x^2 + 3x - 4 = 0, giving (x + 4)(x - 1) = 0, so x = -4 or x = 1. Since the logarithm requires x > 0, the valid solution is x = 1.
Example
For log_3 (x) + log_3 (x - 2) = 1, combine to get x(x - 2) = 3, solve to get x = 3 (since x > 2).
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