If log_5 (x) + log_5 (x + 4) = 1, what is the value of x?
Show the answer and explanation
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_5 [x(x + 4)] = 1 implies x(x + 4) = 5, so x^2 + 4x - 5 = 0. Factoring gives (x + 5)(x - 1) = 0, so x = -5 or x = 1. Since logarithms require positive arguments, x = -5 is extraneous. Thus x = 1.
Example
Solve log_2 (y) + log_2 (y - 2) = 3: y(y - 2) = 8, y^2 - 2y - 8 = 0, (y - 4)(y + 2) = 0, y = 4 (y = -2 extraneous).
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