Question

Let $m$ be the minimum possible value of $\log _{3}\left(3^{y_{1}}+3^{y_{2}}+3^{y_{3}}\right)$, where $y_{1}, y_{2}, y_{3}$ are real numbers for which $y_{1}+y_{2}+y_{3}=9$ .Let $M$ be the maximum possible value of $\left(\log _{3} x_{1}+\log _{3} x_{2}+\log _{3} x_{3}\right)$, where $x_{1}, x_{2}, x_{3}$ are positive real numbers for which $x_{1}+x_{2}+x_{3}=9$ .Then the value of $\log _{2}\left(m^{3}\right)+\log _{3}\left(M^{2}\right)$ is_______

Answer 1

$\text{The Value of}$ $\log_2(m^3)+\log_3(M^2)$ $is\;\underline{8}.$