$$ \begin{aligned} & \log _e a, \log _e b, \log _e c \text { are in A.P. } \\ & \therefore \mathrm{b}^2=\mathrm{ac} \end{aligned} $$ Also $$ \begin{aligned} & \log _e\left(\frac{a}{2 b}\right), \log _e\left(\frac{2 b}{3 c}\right), \log _e\left(\frac{3 c}{a}\right) \text { are in A.P. } \\ & \left(\frac{2 b}{3 c}\right)^2=\frac{a}{2 b} \times \frac{3 c}{a} \\ & \frac{\mathrm{~b}}{\mathrm{c}}=\frac{3}{2} \end{aligned} $$ Putting in eq. (i) $b^2=a \times \frac{2 b}{3}$ $$ \begin{aligned} & \frac{a}{b}=\frac{3}{2} \\ & a: b: c=9: 6: 4 \end{aligned} $$