If log_2 (x) + log_2 (x - 3) = 2, what is the value of x?
Show the answer and explanation
Correct answer
A. 4
Principle or equation
Use the product rule for logarithms: log_b A + log_b B = log_b (AB). Then convert to exponential form and solve the resulting quadratic, checking for domain restrictions.
Why this answer is correct
log_2[x(x-3)] = 2 => x(x-3) = 2^2 = 4 => x^2 - 3x - 4 = 0 => (x-4)(x+1)=0 => x=4 or x=-1. Since log arguments must be positive, x=4 is the only valid solution.
Example
Solve log_3(x) + log_3(x-2) = 1: x(x-2)=3 => x=3 (x=-1 invalid).
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