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Source: 2016 AIME II #3

March 17, 2016
AMCAIMEAIME II

Problem Statement

Let x,yx,y and zz be real numbers satisfying the system \begin{align*} \log_2(xyz-3+\log_5 x) &= 5 \\ \log_3(xyz-3+\log_5 y) &= 4 \\ \log_4(xyz-3+\log_5 z) &= 4. \end{align*} Find the value of log5x+log5y+log5z|\log_5 x|+|\log_5 y|+|\log_5 z|.