Given:

a, b and c are in GP

∴ b² = ac {property of geometric mean}

Taking log on both sides with base m –

log_m b² = log_m ac

⇒ log_m b² = log_m a + log_m c {using property of log}

⇒ 2log_m b = log_m a + log_m c …equation 1

Note: If three numbers a,b and c are in AP,we can say that –

2b = a + c

As equation 1 matches the form above, So

⇒ log_m a, log_m b and log_m c are in AP. …(1)

Now, applying base changing formula we get

⇒ log_ab =

∴ Applying base change on 1, we get

⇒ are in A.P

Hence, proved