Find max value?

Algebra Level 4

\[ \large a^2 b^3 c^4 \]

Let \(a,b\) and \(c\) be non-negative real numbers satisfying \(a+b+c=9\). If the maximum value of the expression above can be expressed as \(2^x 3^y\), where \(x\) and \(y\) are integers, find \(\log_{10} x^y \).

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