Dear Deepti,
use the relation,
logb(a) = loge(a) / loge(b)
thus you get two more relations between a,b,c,d and these can bedidivided in log to get log(b-d).
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b = a3/2  ;   d= c5/4
let  a1/2 = x   and c1/4 =y
then b = x3 and d = y5
given  a-c = 9     or      x2 - y4 = 9     or          (x-y2)(x+y2)=9
we have to find two factors of 9 which are integers
now ; 9 can be expressed as 9x1  or 3x3
i.e. if we breah 9 as 3x3 then x-y2 = x+y2 which is not possible for any existing integer 
therefore  x-y2 = 1  and x+y2 = 9 
adiing the two equations we get     2x = 10   or x=5
and if x = 5 then y = 1
therefore b = 125   and  d =32
therefore b-d = 83
 
 
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in the explantion  above
(x-y2)(x+y2)=9       we have to find two factors of 9 which are integers
Given that a and c are integers
is it neccessary for   (x-y2 )   and     (x+y2  )                 to be integers???????????????