If all the logarithms are real numbers, the equality log(x+3)+log(x-1) =log(x^2-2x-3) is satisfied for:

(C) all real values of x except x=0

(D) no real values of x except x=0

(E) all real values of x except x=1

1968 AMC 12 Problem 23

Attempt Problem 23 of the 1968 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1968 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

23.

If all the logarithms are real numbers, the equality

\begin{aligned} \log(x+3)+\log(x-1)\\ =\log(x^2-2x-3) \end{aligned}

is satisfied for:

all real values of

x

no real values of

x

all real values of

x

except

x=0

no real values of

x

except

x=0

all real values of

x

except

x=1

Combine the two logarithms on the left

Solve the resulting algebraic equation, then test its domain

Combining logarithms would require

(x+3)(x-1)=x^2-2x-3.

This gives

x=0.

But then

x-1=-1,

so the logarithm on the left is not real. Hence there are no real solutions.

Therefore, the correct answer is B.