Take logs of each expression, using ln(x^a) = a ln(x) etc

ln (p*q) = ln(p) + ln(q):

Given,

x^a = x^b/2z^b/2 = z^c

Taking log on each term –

⇒ …(1)

The equality of the first and third expressions tells us that

⇒ …(2)

The second expression is equal to

⇒

⇒ {using equation 1}

∴

Divide through out by ln x

∴ a = b/2 + ab/2c

⇒ 2ac = bc + ab

Dividing the equation by abc –

⇒

From this are in GP.