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2003 AMC 12B Problems/Problem 17

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Problem

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

\mathrm{(A)}\ -\frac 12 \qquad\mathrm{(B)}\ 0 \qquad\mathrm{(C)}\ \frac 12\qquad\mathrm{(D)}\ \frac 35 \qquad\mathrm{(E)}\ 1

Solution

Since \begin{align*}&\log(xy) +2\log y = 1 \\\log(xy) + \log x = 1 \quad \Longrightarrow \quad &2\log(xy) + 2\log x = 2\en... Summing gives 3\log(xy) + 2\log y + 2\log x = 3 \Longrightarrow 5\log(xy) = 3

Hence \log (xy) = \frac 35 \Rightarrow \mathrm{(D)}.

It is not difficult to find x = 10^{2/5}, y = 10^{1/5}.

See also

2003 AMC 12B (Problems • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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