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Difference between revisions of "2009 AIME II Problems/Problem 2"

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== Solution 2 ==
 
== Solution 2 ==
  
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We know from the first three equations that <math>\log_a27</math> = <math>\log_37</math>, <math>\log_b49</math> = <math>\log_711</math>, and <math>\log_c\sqrt{11}</math> = <math>\log_{11}25</math>. Substituting, we find
+
We know from the first three equations that <math>\log_a27 = \log_37</math>, <math>\log_b49 = \log_711</math>, and <math>\log_c\sqrt{11} = \log_{11}25</math>. Substituting, we find
  
−
<math>a^{(\log_a27)(\log_37)} + b^{(\log_b49)(\log_711)} + c^{(\log_c\sqrt {11})(\log_{11}25)}</math>.
+
<cmath>a^{(\log_a27)(\log_37)} + b^{(\log_b49)(\log_711)} + c^{(\log_c\sqrt {11})(\log_{11}25)}.</cmath>
  
 
We know that <math>x^{\log_xy} =y</math>, so we find
 
We know that <math>x^{\log_xy} =y</math>, so we find
  
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<math>27^{\log_37} + 49^{\log_711} + \sqrt {11}^{\log_{11}25}</math>
+
<cmath>27^{\log_37} + 49^{\log_711} + \sqrt {11}^{\log_{11}25}</cmath>
  
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<math>(3^{\log_37})^3 + (7^{\log_711})^2 + ({11^{\log_{11}25}})^{1/2}</math>.
+
<cmath>(3^{\log_37})^3 + (7^{\log_711})^2 + ({11^{\log_{11}25}})^{1/2}.</cmath>
  
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The <math>3</math> and the <math>\log_37</math> cancel out to make <math>7</math>, and we can do this for the other two terms. We obtain
+
The <math>3</math> and the <math>\log_37</math> cancel to make <math>7</math>, and we can do this for the other two terms. Thus, our answer is
  
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<math>7^3 + 11^2 + 25^{1/2}</math>
+
<cmath>7^3 + 11^2 + 25^{1/2}</cmath>
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<math>= 343 + 121 + 5</math>
+
<cmath>= 343 + 121 + 5</cmath>
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<math>= \boxed {469}</math>.
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<cmath>= \boxed {469}.</cmath>
  
 
== See Also ==
 
== See Also ==

Revision as of 18:01, 2 June 2023

Problem

Suppose that $a$, $b$, and $c$ are positive real numbers such that $a^{\log_3 7} = 27$, $b^{\log_7 11} = 49$, and $c^{\log_{11}25} = \sqrt{11}$. Find \[a^{(\log_3 7)^2} + b^{(\log_7 11)^2} + c^{(\log_{11} 25)^2}.\]

Solution 1

First, we have: \[x^{(\log_y z)^2} = x^{\left( (\log_y z)^2 \right) } = x^{(\log_y z) \cdot (\log_y z) } = \left( x^{\log_y z} \right)^{\log_y z}\]

Now, let $x=y^w$, then we have: \[x^{\log_y z} = \left( y^w \right)^{\log_y z} = y^{w\log_y z} = y^{\log_y (z^w)} = z^w\]

This is all we need to evaluate the given formula. Note that in our case we have $27=3^3$, $49=7^2$, and $\sqrt{11}=11^{1/2}$. We can now compute:

\[a^{(\log_3 7)^2} = \left( a^{\log_3 7} \right)^{\log_3 7} = 27^{\log_3 7} = (3^3)^{\log_3 7} = 7^3 = 343\]

Similarly, we get \[b^{(\log_7 11)^2} = (7^2)^{\log_7 11} = 11^2 = 121\]

and \[c^{(\log_{11} 25)^2} = (11^{1/2})^{\log_{11} 25} = 25^{1/2} = 5\]

and therefore the answer is $343+121+5 = \boxed{469}$.

Solution 2

We know from the first three equations that $\log_a27 = \log_37$, $\log_b49 = \log_711$, and $\log_c\sqrt{11} = \log_{11}25$. Substituting, we find

\[a^{(\log_a27)(\log_37)} + b^{(\log_b49)(\log_711)} + c^{(\log_c\sqrt {11})(\log_{11}25)}.\]

We know that $x^{\log_xy} =y$, so we find

\[27^{\log_37} + 49^{\log_711} + \sqrt {11}^{\log_{11}25}\]

\[(3^{\log_37})^3 + (7^{\log_711})^2 + ({11^{\log_{11}25}})^{1/2}.\]

The $3$ and the $\log_37$ cancel to make $7$, and we can do this for the other two terms. Thus, our answer is

\[7^3 + 11^2 + 25^{1/2}\] \[= 343 + 121 + 5\] \[= \boxed {469}.\]

See Also

2009 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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