Question

Which of the following shows the extraneous solution to the logarithmic equation?

log Subscript 4 Baseline (x) + log Subscript 4 Baseline (x minus 3) = log Subscript 4 Baseline (negative 7 x + 21)
x = negative 7
x = negative 3
x = 3 and x = negative 7
x = 7 and x = negative 3

Answer 1

Combine the logs: log_4(x) + log_4(x−3) = log_4(x(x−3)) = log_4(−7x+21).

Exponentiate: x(x−3) = −7x+21 → x^2 +4x −21 = 0 → (x−3)(x+7)=0, so x = 3 or x = −7.

Check domains: log_4(x) and log_4(x−3) require x>3, while log_4(−7x+21) requires x<3. Thus neither x=3 (makes x−3=0) nor x=−7 (x negative) yields valid logs. Both are extraneous.

Answer: x = 3 and x = −7.