Difference between revisions of "2022 AMC 12A Problems/Problem 14"
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What is the value of <cmath>(\log 5)^{3}+(\log 20)^{3}+(\log 8)(\log 0.25)</cmath> where <math>\log</math> denotes the base-ten logarithm? | What is the value of <cmath>(\log 5)^{3}+(\log 20)^{3}+(\log 8)(\log 0.25)</cmath> where <math>\log</math> denotes the base-ten logarithm? | ||
| − | <math>\textbf{(A)}~ | + | <math>\textbf{(A)}~\frac{3}{2}\qquad\textbf{(B)}~\frac{7}{4}\qquad\textbf{(C)}~2\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~3</math> |
==Solution== | ==Solution== | ||
Revision as of 22:06, 11 November 2022
Problem
What is the value of ![\[(\log 5)^{3}+(\log 20)^{3}+(\log 8)(\log 0.25)\]](http://latex.artofproblemsolving.com/a/f/1/af1508c28c6d3e8f63ecd5c43998464209b05c65.png)
where 
denotes the base-ten logarithm?
Solution
Let 
. The expression then becomes ![\[(1+x)^3+(1-x)^3+(3x)(-2x)=\boxed{2}.\]](http://latex.artofproblemsolving.com/2/2/3/223a30976aa281499351492cd728483a4e1456b3.png)
-bluelinfish
See Also
| 2022 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 13 |
Followed by Problem 15 |
| 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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
