Example 7 (2003 National High School Competition Question) Given that $a, b, c, d$ are all positive integers, and $\log _{a} b=\frac{3}{2}, \log _{c} d=\frac{5}{4}$, if $a-c=$ 9, then $b-d=$ $\qquad$ .
Solve: Fill in 93. Reason: Convert the logarithmic mode conditions into exponential mode conditions, we have $a^{\frac{3}{2}}=b, c^{\frac{5}{4}}=d$, which means $a=\left(\frac{b}{a}\right)^{2}, c=\left(\frac{d}{c}\right)^{4}$. Therefore, $a|b, c| d$. Also, since $a-c=9$, it is easy to obtain the model
$$
\begin{array}{l}
\left(\frac{b}{a}+\frac{d^{2}}{c^{2}}\right)\left(\frac{b}{a}-\frac{d^{2}}{c^{2}}\right)=9 . \text { Thus } \\
\frac{b}{a}+\frac{d^{2}}{c^{2}}=9, \\
\frac{b}{a}-\frac{d^{2}}{c^{2}}=1 .
\end{array} \text { Therefore }\left\{\begin{array}{l}
\frac{b}{a}=5, \\
\frac{d^{2}}{c^{2}}=4 .
\end{array}\right.
$$

From $\frac{b}{a}=5$ and $a^{\frac{3}{2}}=b$, we solve to get $a=25$, thus $b=125, c=16, d=32$.