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| − | ==Problem==
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| − | What is the value of <math>\log_2({1+\sqrt{2}+\sqrt{3}})+\log_2({1+\sqrt{2}-\sqrt{3}})</math>?
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| − | <math>\textbf{(A)}~1\qquad\textbf{(B)}~\frac{3}{2}\qquad\textbf{(C)}~2\qquad\textbf{(D)}~\frac{5}{2}\qquad\textbf{(E)}~3</math>
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| − | ==Solution==
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| − | By log properties, we have <math>\log_2({1+\sqrt{2}+\sqrt{3}})+\log_2({1+\sqrt{2}-\sqrt{3}}) = \log_2({(1 + \sqrt{2})^{2} - 3})</math> because of difference of squares. Next, we need to simplify <math>\log_2({1 + 2\sqrt{2} + 2 - 3}) = \log_2{2^{\frac{3}{2}}} = \frac{3}{2}</math> hence <math>\frac{3}{2}</math> is the answer.
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| − | ==See also==
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| − | {{AMC12 box|year=2025|ab=B|before=First Problem|num-a=2}}
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| − | {{MAA Notice}}
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Latest revision as of 11:58, 3 August 2025
Please do not post false problems.
