$\text{ Calculate the value of }\mathrm{x}\text{ from }\frac{\log \left(324\right)}{\log \left(18\right)}=\log \left(\mathrm{x}\right)\text{. }$

$\frac{\log \left(324\right)}{\log \left(18\right)}=\log \left(x\right)$

$$\begin{gathered} \Rightarrow \frac{\log \left(324\right)}{\log \left(18\right)}=\frac{{\log }_{10}\left(324\right)}{{\log }_{10}\left(18\right)} \\ \Rightarrow \frac{{\log }_{10}\left(324\right)}{{\log }_{10}\left(18\right)}={\log }_{10}\left(x\right) \\ \mathrm{By} \mathrm{using} \mathrm{rule}, {\mathrm{a}}^{\mathrm{b}}=\mathrm{x}\iff {\log }_{\mathrm{a}}\mathrm{x}=\mathrm{b}, \mathrm{we} \mathrm{have}: \\ \begin{array}{r}{10}^{\frac{{\log }_{10}\left(324\right)}{{\log }_{10}\left(18\right)}}=\mathrm{x}\\ {10}^{\frac{{\log }_{10}\left({18}^2\right)}{{\log }_{10}\left(18\right)}}=\mathrm{x}\end{array} \\ \mathrm{By} \mathrm{the} \mathrm{power} \mathrm{rule}, \mathrm{we} \mathrm{have}: \\ \begin{array}{r}{10}^{\frac{2{\log }_{10}\left(18\right)}{{\log }_{10}\left(18\right)}}=\mathrm{x}\\ \Rightarrow \mathrm{x}={10}^2\\ \Rightarrow \mathrm{x}=100\end{array} \end{gathered}$$