$$\frac { \log _ { 10 } a } { 2 } = \frac { \log _ { 10 } b } { 3 } = \frac { \log _ { 10 } c } { 5 } e ^ { m } n$$
$\frac{\log(a)}{2}=\frac{\log(b)}{3}\text{ and }\frac{\log(b)}{3}=\frac{n\log(c)e^{m}}{5}$
$\left\{\begin{matrix}n=\frac{5\log_{c}\left(b\right)}{3e^{m}}\text{, }&a=b^{\frac{2}{3}}\text{ and }b>0\text{ and }c\neq 1\text{ and }c>0\\n\in \mathrm{R}\text{, }&b=1\text{ and }c=1\text{ and }a=1\end{matrix}\right.$
