2025 AMC 12B Problems/Problem 1

Problem

What is the value of $\log_2({1+\sqrt{2}+\sqrt{3}})+\log_2({1+\sqrt{2}-\sqrt{3}})$ ?

$\textbf{(A)}~1\qquad\textbf{(B)}~\frac{3}{2}\qquad\textbf{(C)}~2\qquad\textbf{(D)}~\frac{5}{2}\qquad\textbf{(E)}~3$

Solution

By log properties, we have $\log_2({1+\sqrt{2}+\sqrt{3}})+\log_2({1+\sqrt{2}-\sqrt{3}}) = \log_2({(1 + \sqrt{2})^{2} - 3})$ because of difference of squares. Next, we need to simplify $\log_2({1 + 2\sqrt{2} + 2 - 3}) = \log_2{2^{\frac{3}{2}}} = \frac{3}{2}$ hence $\frac{3}{2}$ is the answer.

2025 AMC 12B (Problems • Answer Key • Resources)
Preceded by First Problem	Followed by Problem 2
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All AMC 12 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.

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2025 AMC 12B Problems/Problem 1

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