If log_4 (x) + log_4 (x - 6) = 2, what is the value of x?
Show the answer and explanation
Correct answer
A. 8
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 domain restrictions.
Why this answer is correct
log_4 [x(x - 6)] = 2 => x(x - 6) = 4^2 = 16 => x^2 - 6x - 16 = 0 => (x - 8)(x + 2) = 0 => x = 8 or x = -2. Since logarithms require x > 0 and x - 6 > 0, x > 6, so x = -2 is extraneous. Thus x = 8.
Example
Solve log_2 (x) + log_2 (x - 2) = 3. Combine: log_2 [x(x-2)] = 3 => x(x-2) = 8 => x^2 - 2x - 8 = 0 => (x-4)(x+2)=0 => x=4 (since x>2).
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