2010 AIME II Problems/Problem 5
Problem 5
Positive numbers , , and satisfy and . Find .
Solution
Using the properties of logarithms, by taking the log base 10 of both sides, and by using the fact that . Through further simplification, we find that . It can be seen that there is enough information to use the formula , as we have both and , and we want to find . After plugging in the values into the equation, we find that $\(\log_{10}x)^2 + (\log_{10}y)^2 + (\log_{10}z)^2$ (Error compiling LaTeX. Unknown error_msg) is equal to . However, we want to find , so we take the square root of , or .