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Q.
Let a, b, c, d are positive integers such that $\log _a b=\frac{3}{2}$ and $\log _c d=\frac{5}{4}$. If $(a-c)=9$, find the value of $(b-d)$.
Continuity and Differentiability
Solution:
$b = a ^{3 / 2}$ and $d = c ^{5 / 4}$
let $a=x^2$ and $c=y^4, x, y \in N$
$b = x ^3 ; d = y ^5$
given $a - c =9$
$x^2-y^4=9$
$\left(x-y^2\right)\left(x+y^2\right)=9 ;$ Hence $x-y^2=1$ and $x+y^2=9$
(no other combination in the set of $+v e$ integers will be possible)
$x=5$ and $y=2$
$\therefore b- d = x ^3- y ^5=125-32=93$