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
.