Question

If log₂x = a  and log₃ y = a, write 72^a in terms of x and y.

Answer 1

Given that :
log₂x = a and log₃y = a 
⇒ 2^a = x and 3^a = y       ....[ Q log_am = n ⇒  aⁿ = m ]
Now prime factorization of 72 is 
72 = 2 x 2 x 2 x 3 x 3
Hence,
(72)^a = (2 x 2  x 2 x 3 x 3)^a
        = (2³ x 3²)^a 
       = 2^3a x 3^2a 
       =  (2^a)³ x (3^a)²        ....[ as 2^a = x , 3^a = y ]
       =  x³y²