Given:
a, b and c are in GP
b2 = ac {property of geometric mean}
Apply log on both sides with base m
logm b2 = logm ac
logm b2 = logm a + logm c {using property of log}
2logm b = logm a + logm c
2/logb m = 1/loga m + 1/logc m
∴ 1/loga m , 1/logb m, 1/logc m are in A.P.