2025 AMC 12B Problems/Problem 1: Difference between revisions

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-==Problem==
-What is the value of <math>\log_2({1+\sqrt{2}+\sqrt{3}})+\log_2({1+\sqrt{2}-\sqrt{3}})</math>?
-<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>
-==Solution==
-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.
-==See also==
-{{AMC12 box|year=2025|ab=B|before=First Problem|num-a=2}}
-{{MAA Notice}}

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