Difference between revisions of "2025 AMC 12B Problems/Problem 1"
(Created page with "==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...") |
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==Solution== | ==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. | ||
==See also== | ==See also== | ||
{{AMC12 box|year=2025|ab=B|before=First Problem|num-a=2}} | {{AMC12 box|year=2025|ab=B|before=First Problem|num-a=2}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 23:50, 28 July 2025
Problem
What is the value of 
?
Solution
By log properties, we have 
because of difference of squares. Next, we need to simplify 
hence 
is the answer.
See also
| 2025 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by First Problem |
Followed by Problem 2 |
| 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 | |
| All AMC 12 Problems and Solutions | |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
