2022 AMC 12A Problems/Problem 14
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 1
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
Solution 2
Using sum of cubes
![\[(\log 5)^{3}+(\log 20)^{3}\]](http://latex.artofproblemsolving.com/0/8/a/08a4319f2159439f5ad5508c8c5de2aa8e8dba6c.png)
![\[= (\log 5 + \log 20)((\log 5)^{2}-(\log 5)(\log 20) + (\log 20)^{2})\]](http://latex.artofproblemsolving.com/2/0/7/2074325451a4f90d6d79749f3e1d47a8eaa84abe.png)
![\[= 2((\log 5)^{2}-(\log 5)(2\log 2 + \log 5) + (2\log 2 + \log 5)^{2})\]](http://latex.artofproblemsolving.com/7/d/e/7de8d5ee2cfb76914cd7099210040eb9b53dd8d4.png)
Let x = 
and y = 
, so 
The entire expression becomes
![\[2(x^2-x(2y+x)+(2y+x)^2)-6y^2\]](http://latex.artofproblemsolving.com/5/e/4/5e46b1000679ebea29bea0016de5c5f3f1c28b16.png)
![\[=2(x^2+2xy+4y^2-3y^2)\]](http://latex.artofproblemsolving.com/1/b/2/1b2479a5de4184f7d0a9028cb3a8c19ebd3073dd.png)
![\[=2(x+y)^2 = \boxed{2}\]](http://latex.artofproblemsolving.com/7/c/8/7c83b8d9cc53bb9158a7d62ec9a1eed6a878138e.png)
~Hithere22702
Solution 3
We can estimate the solution. Using 
and 
we have
![\[0.7^3 + 1.7^3 + ).9\cdot(-0.6) = \boxed{2}\]](http://latex.artofproblemsolving.com/a/b/e/abe50cb2b8725fd3ae63c24a55fb9b72b55284ed.png)
~kxiang
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 | |
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