If log_3 (x) + log_3 (x - 2) = 1, what is the value of x?
Show the answer and explanation
Correct answer
A. 3
Principle or equation
Use the logarithm product rule: log_b A + log_b B = log_b (AB). Then convert to exponential form.
Why this answer is correct
log_3 [x(x - 2)] = 1 ⇒ x(x - 2) = 3^1 = 3 ⇒ x^2 - 2x - 3 = 0 ⇒ (x - 3)(x + 1) = 0 ⇒ x = 3 or x = -1. Since logarithms require x > 0 and x - 2 > 0, x must be greater than 2, so x = 3.
Example
Solve log_2 (x) + log_2 (x - 1) = 1. Then x(x - 1) = 2, giving x = 2 or x = -1; only x = 2 is valid.
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