\(m=\frac{1}{3}log_a\left(ab\right)=\frac{1}{3}+\frac{1}{3}log_ab\)
\(\Rightarrow log_ab=3m-1>0\)
\(P=\left(log_ab\right)^2+\frac{16}{log_ab}=\left(3m-1\right)^2+\frac{16}{3m-1}\)
\(P=\left(3m-1\right)^2+\frac{8}{3m-1}+\frac{8}{3m-1}\ge3\sqrt[3]{\frac{64\left(3m-1\right)^2}{\left(3m-1\right)^2}}=12\)
\(P_{min}=12\) khi \(\left(3m-1\right)^3=8\Rightarrow m=1\)