Question

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 $(b-d)$.

Answer 1

Answer: 93

$b=a^{3/2}$ 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 $+ve$ integers will be possible)
$x=5$ and $y=2$
$\therefore b-d=x^3-y^5=125-32=93$