If log_5 (x) + log_5 (x - 4) = 1, what is the value of x?
Show the answer and explanation
Correct answer
A. 5
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 equation, checking for domain restrictions.
Why this answer is correct
Combine logs: log_5[x(x - 4)] = 1. Then x(x - 4) = 5^1 = 5, so x^2 - 4x - 5 = 0. Factoring gives (x - 5)(x + 1) = 0, so x = 5 or x = -1. Since logarithms require x > 0 and x - 4 > 0, x > 4, so x = -1 is extraneous. Therefore x = 5.
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 is invalid).
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