2022 AMC 12A Problems/Problem 14: Difference between revisions

Latest revision as of 14:24, 5 August 2025

Problem

What is the value of $(\log 5)^{3}+(\log 20)^{3}+(\log 8)(\log 0.25)$ where $\log$ denotes the base-ten logarithm?

$\textbf{(A)}~\frac{3}{2}\qquad\textbf{(B)}~\frac{7}{4}\qquad\textbf{(C)}~2\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~3$

Solution 1

Let $\text{log } 2 = x$ . The expression then becomes $(1+x)^3+(1-x)^3+(3x)(-2x)=\boxed{2}.$

-bluelinfish

Solution 1.1

(Elaboration & motivation behind Sol. 1)

Note that $5,20,8$ , and $0.25$ are all products and quotients of exponents of $2$ and $5$ , and the base of the logarithms is $10 = 2\times5$ ; this strongly hints at some sort of major simplification using the addition and subtraction rules of logarithms so we can convert all the different arguments of the logs into 1 common argument for easy algebra.

Note that we can write all of the following expressions in the following ways:

\begin{align*} \log5=\log\dfrac{10}2=\log10-\log2&=1-\log2\\ \log20=\log(2\cdot10)=\log2+\log10&=\log2+1\\ \log8=\log\left(2^3\right)&=3\log2 \\ \log0.25=\log\left(2^{-2}\right)&=-2\log2 \end{align*}

Thus, let $a=\log2$ , and proceed as in solution 1.

~Technodoggo ~some elaboration by rawr3507

Solution 2

Using sum of cubes $(\log 5)^{3}+(\log 20)^{3}$ $= (\log 5 + \log 20)((\log 5)^{2}-(\log 5)(\log 20) + (\log 20)^{2})$ $= 2((\log 5)^{2}-(\log 5)(2\log 2 + \log 5) + (2\log 2 + \log 5)^{2})$ Let x = $\log 5$ and y = $\log 2$ , so $x+y=1$

The entire expression becomes $2(x^2-x(2y+x)+(2y+x)^2)-6y^2$ $=2(x^2+2xy+4y^2-3y^2)$ $=2(x+y)^2 = \boxed{2}$

~kempwood

Solution 3 (Estimates)

We can estimate the solution. Using $\log(2) \approx 0.3, \log(20) = \log(10)+\log(2) = 1 + 0.3 \approx 1.3, \log(8) \approx 0.9$ and $\log(.25) = \log(1)-\log(4)= 0 - 0.6\approx -0.6,$ we have

$0.7^3 + 1.3^3 + .9\cdot(-0.6) = \boxed{2}$ ~kxiang

Solution 4(log bash)

Using log properties, we combine the terms to make our expression equal to $\log {( (5^{\log^2{5}}) \cdot (20^{\log^2{20}}) \cdot 8 ^ {\log{\frac{1}{4}}} ) }$ . By exponent properties, we separate the part with base $20$ to become $20^{\log^2{5}} \cdot 20^{\log^2{20}-\log^2{5}}$ . Then, we substitute this into the original expression to get $\log {( (5^{\log^2{5}}) \cdot 20^{\log^2{5}} \cdot 20^{\log^2{20}-\log^2{5}} \cdot 8 ^ {\log{\frac{1}{4}}} ) } = \log {( (100^{\log^2{5}}) \cdot 20^{\log^2{20}-\log^2{5}} \cdot 8 ^ {\log{\frac{1}{4}}} ) }$ . Because $100^{\log^2{5}} = 25^{\log{5}}$ , and $\log^2{20}-\log^2{5} = (\log{20}+\log{5})(\log{20}-\log{5}) = \log{100}\cdot\log{4} = 2\log{4}$ , this expression is equal to $\log {( 25 ^ {\log{5}} \cdot 400^{\log{4}} \cdot 8 ^ {\log{\frac{1}{4}}} ) }$ . We perform the step with the base combining on $25$ and $400$ to get $25 ^ {\log{5}} \cdot 400^{\log{4}} = 25 ^ {\log{5}-\log{4}} \cdot 10000^{\log{4}} = 25^{\log{\frac{5}{4}}}\cdot 256$ . Putting this back into the whole equation gives $\log{( 25^{\log{\frac{5}{4}}}\cdot 256 \cdot 8^{\log{\frac{1}{4}}})}$ . One last base merge remains - but $25\cdot 8$ isn't a power of 10. We can rectify this by converting $8^{\log{\frac{1}{4}}}$ to $(4^\frac{3}{2})^{\log{\frac{1}{4}}} = 4^{\log{ \frac{1}{8} }}$ . Finally, we complete this arduous process by performing the base merge on $\log{( 25^{\log{\frac{5}{4}}}\cdot 256 \cdot 4^{\log{\frac{1}{8}}})}$ . We get $25^{\log{\frac{5}{4}}} \cdot 4^{\log{\frac{1}{8}}} = 25^{\log{\frac{5}{4}}-\log{\frac{1}{8}}} \cdot 100^{\log{\frac{1}{8}}} = 25^{\log{10}} \cdot \frac{1}{64} = \frac{25}{64}$ . Putting this back into that original equation one last time, we get $\log(256 \cdot \frac{25}{64}) = \log{100} = \boxed{2}$ . ~aop2014

Video Solution (Speedy)

https://www.youtube.com/watch?v=pai2A9FXI9U

~Education, the Study of Everything

Video Solution (Simple)

https://youtu.be/7yAh4MtJ8a8?si=9vbP5erdxlCLlG82&t=2957

~Math-x

Video Solution (Most Effecient)

https://www.youtube.com/watch?v=ndtpVFADLUo

@@ Line 4: / Line 4: @@
 <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 1==
 Let <math>\text{log } 2 = x</math>. The expression then becomes <cmath>(1+x)^3+(1-x)^3+(3x)(-2x)=\boxed{2}.</cmath>
 -bluelinfish
+===Solution 1.1===
+(Elaboration & motivation behind Sol. 1)
+Note that <math>5,20,8</math>, and <math>0.25</math> are all products and quotients of exponents of <math>2</math> and <math>5</math>, and the base of the logarithms is <math>10 = 2\times5</math>; this strongly hints at some sort of major simplification using the addition and subtraction rules of logarithms so we can convert all the different arguments of the logs into 1 common argument for easy algebra.
+Note that we can write all of the following expressions in the following ways:
+\begin{align*}
+\log5=\log\dfrac{10}2=\log10-\log2&=1-\log2\\ \log20=\log(2\cdot10)=\log2+\log10&=\log2+1\\
+\log8=\log\left(2^3\right)&=3\log2 \\
+\log0.25=\log\left(2^{-2}\right)&=-2\log2
+\end{align*}
+Thus, let <math>a=\log2</math>, and proceed as in solution 1.
+~Technodoggo
+~some elaboration by rawr3507
+==Solution 2==
+Using sum of cubes
+<cmath>(\log 5)^{3}+(\log 20)^{3}</cmath>
+<cmath>= (\log 5 + \log 20)((\log 5)^{2}-(\log 5)(\log 20) + (\log 20)^{2})</cmath>
+<cmath>= 2((\log 5)^{2}-(\log 5)(2\log 2 + \log 5) + (2\log 2 + \log 5)^{2})</cmath>
+Let x = <math>\log 5</math> and y = <math>\log 2</math>, so <math>x+y=1</math>
+The entire expression becomes
+<cmath>2(x^2-x(2y+x)+(2y+x)^2)-6y^2</cmath>
+<cmath>=2(x^2+2xy+4y^2-3y^2)</cmath>
+<cmath>=2(x+y)^2 = \boxed{2}</cmath>
+~kempwood
+==Solution 3 (Estimates)==
+We can estimate the solution. Using <math>\log(2) \approx 0.3, \log(20) = \log(10)+\log(2) = 1 + 0.3 \approx 1.3, \log(8) \approx 0.9</math> and <math>\log(.25) = \log(1)-\log(4)= 0 - 0.6\approx -0.6,</math> we have
+<cmath>0.7^3 + 1.3^3 + .9\cdot(-0.6) = \boxed{2}</cmath>
+~kxiang
+==Solution 4(log bash)==
+Using log properties, we combine the terms to make our expression equal to <math>\log {( (5^{\log^2{5}}) \cdot (20^{\log^2{20}}) \cdot 8 ^ {\log{\frac{1}{4}}} ) }</math>.
+By exponent properties, we separate the part with base <math>20</math> to become <math>20^{\log^2{5}} \cdot 20^{\log^2{20}-\log^2{5}}</math>. Then, we substitute this into the original expression to get <math>\log {( (5^{\log^2{5}}) \cdot 20^{\log^2{5}} \cdot 20^{\log^2{20}-\log^2{5}} \cdot 8 ^ {\log{\frac{1}{4}}} ) } = \log {( (100^{\log^2{5}}) \cdot 20^{\log^2{20}-\log^2{5}} \cdot 8 ^ {\log{\frac{1}{4}}} ) }</math>.
+Because <math>100^{\log^2{5}} = 25^{\log{5}}</math>, and <math>\log^2{20}-\log^2{5} = (\log{20}+\log{5})(\log{20}-\log{5}) = \log{100}\cdot\log{4} = 2\log{4}</math>, this expression is equal to <math> \log {( 25 ^ {\log{5}} \cdot 400^{\log{4}} \cdot 8 ^ {\log{\frac{1}{4}}} ) }</math>.
+We perform the step with the base combining on <math>25</math> and <math>400</math> to get <math>25 ^ {\log{5}} \cdot 400^{\log{4}}  = 25 ^ {\log{5}-\log{4}} \cdot 10000^{\log{4}} = 25^{\log{\frac{5}{4}}}\cdot 256</math>. Putting this back into the whole equation gives <math>\log{( 25^{\log{\frac{5}{4}}}\cdot 256 \cdot 8^{\log{\frac{1}{4}}})}</math>.
+One last base merge remains - but <math>25\cdot 8</math> isn't a power of 10. We can rectify this by converting <math>8^{\log{\frac{1}{4}}}</math> to <math>(4^\frac{3}{2})^{\log{\frac{1}{4}}} = 4^{\log{ \frac{1}{8} }}</math>.
+Finally, we complete this arduous process by performing the base merge on <math>\log{( 25^{\log{\frac{5}{4}}}\cdot 256 \cdot 4^{\log{\frac{1}{8}}})}</math>.
+We get <math>25^{\log{\frac{5}{4}}} \cdot 4^{\log{\frac{1}{8}}} = 25^{\log{\frac{5}{4}}-\log{\frac{1}{8}}} \cdot 100^{\log{\frac{1}{8}}} = 25^{\log{10}} \cdot \frac{1}{64} = \frac{25}{64}</math>.
+Putting this back into that original equation one last time, we get <math>\log(256 \cdot \frac{25}{64}) = \log{100} = \boxed{2}</math>.
+~aop2014
+==Video Solution (Speedy)==
+https://www.youtube.com/watch?v=pai2A9FXI9U
+~Education, the Study of Everything
+==Video Solution (Simple)==
+https://youtu.be/7yAh4MtJ8a8?si=9vbP5erdxlCLlG82&t=2957
+~Math-x
+==Video Solution (Most Effecient)==
+https://www.youtube.com/watch?v=ndtpVFADLUo
 ==See Also==
 {{AMC12 box|year=2022|ab=A|num-b=13|num-a=15}}
 {{MAA Notice}}

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

Page

Toolbox

Search