2003 AMC 12B Problem 17 — Solution | LIVE by Po-Shen Loh

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2003 AMC 12B Problem 17

Below is the professionally curated solution for Problem 17 of the 2003 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 12B solutions, or check the answer key.

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Concepts:logarithm system of equations

Difficulty rating: 1540

17.

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

$-\dfrac{1}{2}$

$0$

$\dfrac{1}{2}$

$\dfrac{3}{5}$

$1$

Solution:

Let $X = \log x$ and $Y = \log y.$ Then $X + 3Y = 1 \quad\text{and}\quad 2X + Y = 1.$

Solving gives $X = \dfrac{2}{5}$ and $Y = \dfrac{1}{5},$ so $\log(xy) = X + Y = \frac{3}{5}.$

Thus, the correct answer is D.

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