If log_7 (x) + log_7 (x - 6) = 1, what is the value of x?
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Correct answer
A. 7
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_7 (x) + log_7 (x - 6) = 1 => log_7 [x(x - 6)] = 1 => x(x - 6) = 7^1 = 7 => x^2 - 6x - 7 = 0 => (x - 7)(x + 1) = 0, so x = 7 or x = -1. Since x must be positive and x - 6 > 0, x > 6, so x = 7 is the only valid solution.
Example
Solve log_2 (y) + log_2 (y - 2) = 3. Then y(y - 2) = 8 => y^2 - 2y - 8 = 0 => y = 4 or y = -2, so y = 4.
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