Difference between revisions of "2013 AMC 12B Problems/Problem 22"
(Created page with "==Problem== Let <math>m>1</math> and <math>n>1</math> be integers. Suppose that the product of the solutions for <math>x</math> of the equation <cmath> 8(\log_n x)(\log_m x)-7\lo...") |
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<math> \textbf{(A)}\ 12\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 24\qquad\textbf{(D}}\ 48\qquad\textbf{(E)}\ 272 </math> | <math> \textbf{(A)}\ 12\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 24\qquad\textbf{(D}}\ 48\qquad\textbf{(E)}\ 272 </math> | ||
+ | |||
+ | ==Solution== | ||
+ | Rearranging logs, the original equation becomes | ||
+ | <cmath>\frac{8}{\log n \log m}(\log x)^2 - \left(\frac{7}{\log n}+\frac{6}{\log m}\right)\log x - 2013 = 0</cmath> | ||
+ | |||
+ | By Vieta's Theorem, the sum of the possible values of <math>\log x</math> is <math>\frac{\frac{7}{\log n}+\frac{6}{\log m}}{\frac{8}{\log n \log m}} = \frac{7\log m + 6 \log n}{8} = \log \sqrt[8]{m^7n^6}</math>. But the sum of the possible values of <math>\log x</math> is the logarithm of the product of the possible values of <math>x</math>. Thus the product of the possible values of <math>x</math> is equal to <math>\sqrt[8]{m^7n^6}</math>. | ||
+ | |||
+ | It remains to minimize the integer value of <math>\sqrt[8]{m^7n^6}</math>. Since <math>m, n>1</math>, we can check that <math>m = 2^2</math> and <math>n = 2^3</math> work. Thus the answer is <math>4+8 = \boxed{\textbf{(A)}\ 12}</math>. |
Revision as of 15:53, 22 February 2013
Problem
Let and be integers. Suppose that the product of the solutions for of the equation is the smallest possible integer. What is ?
$\textbf{(A)}\ 12\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 24\qquad\textbf{(D}}\ 48\qquad\textbf{(E)}\ 272$ (Error compiling LaTeX. Unknown error_msg)
Solution
Rearranging logs, the original equation becomes
By Vieta's Theorem, the sum of the possible values of is . But the sum of the possible values of is the logarithm of the product of the possible values of . Thus the product of the possible values of is equal to .
It remains to minimize the integer value of . Since , we can check that and work. Thus the answer is .