If log_6 (x) + log_6 (x + 5) = 2, what is the value of x?
Show the answer and explanation
Correct answer
A. 4
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_6 [x(x+5)] = 2 -> x(x+5) = 6^2 = 36 -> x^2 + 5x - 36 = 0 -> (x+9)(x-4)=0 -> x = -9 or 4. Since logarithms require positive arguments, x = -9 is invalid. Thus x = 4.
Example
Solve log_2 (x) + log_2 (x - 2) = 3. Then x(x-2)=8 -> x^2 -2x -8=0 -> (x-4)(x+2)=0 -> x=4 (x=-2 invalid).
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