$$\log _{\mathrm{e}} \mathrm{a}, \log _{\mathrm{e}} \mathrm{b}, \log _{\mathrm{e}} \mathrm{c}$$ are in A.P.

$$\therefore \mathrm{b}^2=\mathrm{ac}$$ ..... (i)

Also

$$\begin{aligned} & \log _e\left(\frac{a}{2 b}\right), \log _e\left(\frac{2 b}{3 c}\right), \log _e\left(\frac{3 c}{a}\right) \text { are in A.P. } \\ & \left(\frac{2 b}{3 c}\right)^2=\frac{a}{2 b} \times \frac{3 c}{a} \\ & \frac{b}{c}=\frac{3}{2} \end{aligned}$$

Putting in eq. (i) $$b^2=a \times \frac{2 b}{3}$$

$$\begin{aligned} & \frac{\mathrm{a}}{\mathrm{b}}=\frac{3}{2} \\ & \mathrm{a}: \mathrm{b}: \mathrm{c}=9: 6: 4 \end{aligned}$$