Solve for x: log₃(x) + log₃(x - 2) = 1.
Show the answer and explanation
Correct answer
A. x = 3
Principle or equation
Use the logarithm product rule: logₐ(m) + logₐ(n) = logₐ(mn). Then convert to exponential form and solve the resulting quadratic, checking for extraneous solutions.
Why this answer is correct
log₃(x(x - 2)) = 1 ⇒ x(x - 2) = 3¹ ⇒ x² - 2x - 3 = 0 ⇒ (x - 3)(x + 1) = 0 ⇒ x = 3 or x = -1. Since logarithms require positive arguments, x = -1 is invalid. Thus x = 3.
Example
Solve log₂(x) + log₂(x - 1) = 1: x(x - 1) = 2 ⇒ x² - x - 2 = 0 ⇒ x = 2 or x = -1; valid is x = 2.
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