If log_4 (x) + log_4 (x - 3) = 1, 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 domain restrictions.
Why this answer is correct
log_4 [x(x - 3)] = 1 => x(x - 3) = 4^1 = 4 => x^2 - 3x - 4 = 0 => (x - 4)(x + 1) = 0 => x = 4 or x = -1. Since logarithms require x > 0 and x - 3 > 0, x > 3, so x = -1 is extraneous. Thus x = 4.
Example
Solve log_2 x + log_2 (x - 2) = 3: x(x - 2) = 8 => x^2 - 2x - 8 = 0 => x = 4 or x = -2; only x = 4 is valid.
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