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

## 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 $a^{(log_a27)(log_37)}$ + $b^{(log_b49)(log_711)$ (Error compiling LaTeX. ! Missing } inserted.) + $c^{(log_c\sqrt {11})(log_{11}25)}$

We know that $x^{log_xy}$ = $y$, so we get $27^{log_37}$ + $49^{log_711}$ + $\sqrt {11}^{log_{11}25}$ $(3^{log_37})^3$ + $(7^{log_711})^2$ + ${11^{log_{11}25}^1/2$ (Error compiling LaTeX. ! Double superscript.)

The $3$ and the $log_37$ cancel out to make $7$, and we can do this for the other two terms. We obtain $7^3$ + $11^2$ + $25^{1/2}$

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