Verify if the sequence where loga, log(a^2/b), log(a^3/b^2), ... is an A.P.

We'll have at least 2 methods to prove
that.

We'll verify if the difference between 2 consecutive
terms of the sequence is the same.

We'll note the
consecutive terms as t1, t2, t3, where:

t1 = log
a

t2 = log(a^2/b)

t3 =
log(a^3/b^2)

We'll calculate the difference between t2 and
t1:

t2 - t1 = loga -
log(a^2/b)

We'll use the quotient property of the
logarithms:

loga - log(a^2/b) = log
(a*b/a^2)

We'll eliminate like
terms:

log (a*b/a^2) = log
(b/a)

t2 - t1 = log
(b/a)

We'll calculate the difference between t3 and
t2:

t3 - t2 = log(a^3/b^2) -
log(a^2/b)

We'll use the quotient property of the
logarithms, once again:

t3 - t2 = log (a^3 * b/b^2 *
a^2)

We'll eliminate like
terms:

t3 - t2 = log (a/b)

We
notice that the difference between t2 and t1, t3 and t2 and so on is the same quantity:
log (a/b).

So, the difference is the common difference
between 2 consecutive terms of the sequence and the sequence is an Arithmetical
Progression.