Subject 4. Let $a, b, c, d \geq 2$ be natural numbers such that

$$
\log _{a} b=\frac{3}{2}, \quad \log _{c} d=\frac{5}{4}
$$

and $a-c=9$. Calculate $b-d$.

Mathematical Gazette 2013

## GRADING SCALE

10th GRADE
Subject 4. $a^{3 / 2}=b, c^{5 / 4}=d(\mathbf{1} \mathbf{p}) ; a^{3}=b^{2}, c^{5}=d^{4}(\mathbf{1} \mathbf{p}) ; a=x^{2}, b=x^{3}$ $(\mathbf{1} \mathbf{p}) ; c=y^{4}, d=y^{5}(\mathbf{1} \mathbf{p}) ; a-c=x^{2}-y^{4}=9(\mathbf{1} \mathbf{p}) ;\left(x-y^{2}\right)\left(x+y^{2}\right)=$ $9 \Rightarrow x=5, y=2(\mathbf{1} \mathbf{p}) ; b-d=x^{3}-y^{5}=93(\mathbf{1} \mathbf{p})$.