The number of roots of the equation ${\log }_{3\sqrt{\mathrm{x}}}\mathrm{x}+{\log }_{3\mathrm{x}}\sqrt{\mathrm{x}}=0$

$\begin{array}{r}\frac{\log \mathrm{x}}{\log 3+(1/2)\log \mathrm{x}}+\frac{(1/2)\log \mathrm{x}}{\log 3+\log \mathrm{x}}=0\\ \Rightarrow \frac{{\log }_3\mathrm{x}}{1+(1/2){\log }_3\mathrm{x}}+\frac{1}{2}\frac{{\log }_3\mathrm{x}}{\left(1+{\log }_3\mathrm{x}\right)}=0\end{array}$

$\text{ Let }{\log }_3\mathrm{x}=\mathrm{y}\text{, we get }\frac{\mathrm{y}}{1+(\mathrm{y}/2)}+\frac{\mathrm{y}}{2(1+\mathrm{y})}=0$

$\begin{array}{l}\text{ or }\mathrm{y}\left(\frac{2}{2+\mathrm{y}}+\frac{1}{2(1+\mathrm{y})}\right)=0\\ \text{ or }\mathrm{y}[4+4\mathrm{y}+2+\mathrm{y}]=0\\ \Rightarrow \mathrm{y}=0\text{ or }\mathrm{y}=-6/5\\ \Rightarrow {\log }_3\mathrm{x}=0\text{ or }{\log }_3\mathrm{x}=-6/5\\ \Rightarrow \mathrm{x}=1\text{ or }\mathrm{x}=3^{-6/5}\end{array}$