AoPSWiki

Art of Problem Solving holds many free classes called Math Jams.
Click here for transcripts to past Math Jams.

Personal tools

Search

Toolbox

2009 AIME II Problems/Problem 2

From AoPSWiki

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

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 get

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

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

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

$(3^{log_37})^3$ + $(7^{log_711})^2$ + $({11^{log_{11}25})^{1/2}$

The $3$ and the $log_37$ cancel out to make $7$ , and we can do this for the other two terms. We obtain

$7^3$ + $11^2$ + $25^{1/2}$

= $343$ + $121$ + $5$ = $\boxed {469}$ .

See Also

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

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/2009_AIME_II_Problems/Problem_2"

Try our innovative online adaptive learning system, Alcumus.
Over 1100 problems and 60+ video lessons. FREE!

© Copyright 2008 AoPS Incorporated. All Rights Reserved. • Foundation • Privacy • Contact Us