2022-07-14

Find $b-d$ when ${\mathrm{log}}_{a}b=\frac{3}{2}$ and ${\mathrm{log}}_{c}d=\frac{5}{4}$
$a,b,c$ are three natural numbers such that ${\mathrm{log}}_{a}b=\frac{3}{2}$ and ${\mathrm{log}}_{c}d=\frac{5}{4}$. Given: $a-c=9$
Find $b-d$

yermarvg

Expert

Hint:

Since $\alpha +{\beta }^{2}>0$ the other factor of $9$ ($\alpha -{\beta }^{2}$) must be $>0⇒$
$\left\{\begin{array}{l}\alpha +{\beta }^{2}=1,\alpha -{\beta }^{2}=9\\ \alpha +{\beta }^{2}=3,\alpha -{\beta }^{2}=3\\ \alpha +{\beta }^{2}=9,\alpha -{\beta }^{2}=1\end{array}$

The only solution is
$a=25\phantom{\rule{2em}{0ex}}b=125\phantom{\rule{2em}{0ex}}c=16\phantom{\rule{2em}{0ex}}d=32$
$⇒b-d=125-32=93$

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