223. $\log x=\frac{1}{3}(\log a+\log b)-\frac{1}{2} \log (a+c)$.
Solution. Using the converse statements of Theorems 1, 3, 4, and 2, we have

$$
\begin{gathered}
\log x=\frac{1}{3} \log (a b)-\frac{1}{2} \log (a+c)=\log \sqrt[3]{a b}-\log \sqrt{a+c}=\log \frac{\sqrt[3]{a b}}{\sqrt{a+c}} ; \\
\log x=\log \frac{\sqrt[3]{a b}}{\sqrt{a+c}} ; \quad x=\frac{\sqrt[3]{a b}}{\sqrt{a+c}}
\end{gathered}
$$