19. Let $x$ be a positive real number. What is the maximum value of $\frac{2022 x^{2} \log (x+2022)}{(\log (x+2022))^{3}+2 x^{3}}$ ?
19. 674
By the AM-GM Inequality,
$$
\begin{aligned}
\frac{2022 x^{2} \log (x+2022)}{\log ^{3}(x+2022)+2 x^{3}} & =\frac{2022 x^{2} \log (x+2022)}{\log ^{3}(x+2022)+x^{3}+x^{3}} \\
& \leq \frac{2022 x^{2} \log (x+2022)}{3\left(x^{6} \log ^{3}(x+2022)\right)^{\frac{1}{3}}} \\
& =\frac{2022 x^{2} \log (x+2022)}{3 x^{2} \log (x+2022)} \\
& =674
\end{aligned}
$$

Equality holds when $x=\log (x+2022)$, which has a positive solution.