If log_3 (x) + log_3 (x + 8) = 2, what is the value of x?
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Correct answer
A. 1
Principle or equation
Use the logarithm product rule: log_b A + log_b B = log_b (AB). Then convert to exponential form and solve the resulting quadratic, checking for extraneous solutions.
Why this answer is correct
log_3 [x(x+8)] = 2 → x(x+8) = 3^2 = 9 → x^2 + 8x - 9 = 0 → (x+9)(x-1)=0 → x = -9 or x = 1. Since x must be positive, x = 1.
Example
Solve log_2 (x) + log_2 (x+2) = 3: x(x+2)=8 → x^2+2x-8=0 → x=2 or -4; x=2.
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