If log_2 (x) + log_2 (x - 1) = 1, what is the value of x?
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Correct answer
A. 2
Principle or equation
Use logarithm product rule: log_b A + log_b B = log_b (AB). Then convert to exponential form and solve the quadratic.
Why this answer is correct
log_2 [x(x-1)] = 1 => x(x-1) = 2^1 = 2 => x^2 - x - 2 = 0 => (x-2)(x+1)=0 => x=2 or x=-1. Since log arguments must be positive, x>1, so x=2.
Example
Solve log_3 (x) + log_3 (x-2) = 1: x(x-2)=3 => x^2-2x-3=0 => x=3 or -1, valid x=3.
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