If \(\log _e a , \log _e b , \log _e c\) are in an A.P. and \(\log _e a -\log _e 2 b , \log _e 2 b -\log _e 3 c , \log _…

\( \log _e a, \log _e b, \log _e c \) are in A.P.
\(\therefore \mathrm{b}^2=\mathrm{ac} \quad \ldots .(i)\)

Also \(\log _e\left(\frac{a}{2 b}\right), \log _c\left(\frac{2 b}{3 c}\right), \log _e\left(\frac{3 c}{a}\right)\) are in A.P.

\( \left(\frac{2 \mathrm{~b}}{3 \mathrm{c}}\right)^2=\frac{\mathrm{a}}{2 \mathrm{~b}} \times \frac{3 \mathrm{c}}{\mathrm{a}} \)
\( \frac{\mathrm{~b}}{\mathrm{c}}=\frac{3}{2}\)

Putting in eq. (i)

\( b^2=a \times \frac{2 b}{3} \)
\(\frac{a}{b}=\frac{3}{2} \)
\( a: b: c=9: 6: 4\)