If $a-c = 9$ then find the value of $b-d$. If $a,b,c,d$ are positive integers such that $\log_a b=\frac{3}{2}$ and $\log_c d=\frac{5}{4}$, if $a-c = 9$ then find the value of $b-d$.
We get
$b=a^{3/2}$ and $d=c^{5/4}$
Hence
$b-d=a^{3/2}-c^{5/4}$
Now we need to use the fact that $a,b,c,d$ are integers, but is hit and trial the only way to go here?

$$a=b^\frac{2}{3}$$ 
$$c=d^\frac{4}{5}$$ 
$$a-c=b^\frac{2}{3}-d^\frac{4}{5}$$ 
Clearly $$b=5^3$$ and
$$d=2^5$$ 
satisfies the equation.
