2. All real values of $x$ that satisfy $\lg (x+1)=\frac{1}{2} \log _{3} x$ are $\qquad$
2. 9 Detailed Explanation: Let $\lg (x+1)=\frac{1}{2} \log _{3} x=k$, then $\frac{x}{x+1}=\left(\frac{9}{10}\right)^{k}, \frac{1}{x+1}=\left(\frac{1}{10}\right)^{k},\left(\frac{9}{10}\right)^{k}+\left(\frac{1}{10}\right)^{k}=1$, because $f(k)=\left(\frac{9}{10}\right)^{k}+\left(\frac{1}{10}\right)^{k}$ is decreasing with respect to $k$, and $f(1)=1$, so $k=1, x=9$.