Log!

$\log_ab=\dfrac {3}{2}$ $\Rightarrow b=a^{\frac {3}{2}}=a\sqrt{a}$ $log_cd=\dfrac {5}{4}$ $\Rightarrow d=c^{\frac {5}{4}}=c\sqrt[4]{c}$

Since $b$ and $d$ are both positive integers, both $\sqrt{a}$ and $\sqrt[4]{c}$ must be positive integers as well.

Hence, let $a=x^2,c=y^4$ for some positive integers $x$ and $y$ .

$x^2-y^4=9$ $\Rightarrow (x-y^2)(x+y^2)=9$

Note that $x-y^2 < x+y^2$ so $x-y^2=1$ and $x+y^2=9$ .

Thus $x=5,y=2$ $\Rightarrow b-d = x^3-y^5 = 125-32 = \boxed{93}$