If log(xy^3) = 1 and log(x^2y) = 1, what is log(xy)?

2003 AMC 12B Problem 17

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17.

\log(xy^3) = 1

and

\log(x^2y) = 1,

what is

\log(xy)?

-\dfrac{1}{2}

0

\dfrac{1}{2}

\dfrac{3}{5}

1

Let

X = \log x

and

Y = \log y;

the equations become linear in

X

and

Y

Solve

X + 3Y = 1

and

2X + Y = 1,

then compute

X + Y

Let

X = \log x

and

Y = \log y.

Then

X + 3Y = 1

and

2X + Y = 1.

Solving gives

X = \dfrac{2}{5}

and

Y = \dfrac{1}{5},

\log(xy) = X + Y = \frac{3}{5}.

Thus, the correct answer is D.