What is value of this equation?

Question

$\displaystyle \frac{\log a}{\log b}=\frac{1}{2}$, $\displaystyle \;\frac{\log c}{\log d}=\frac{3}{4}$, $\; a-c=9$.

What is $b-d$ equal to?

Answer 1

HINT:

$$ \begin{cases} \frac{\ln(\text{a})}{\ln(\text{b})}=\frac{1}{2}\\ \frac{\ln(\text{c})}{\ln(\text{d})}=\frac{3}{4}\\ \text{a}-\text{c}=9 \end{cases}\space\space\space\Longleftrightarrow\space\space\space \begin{cases} \ln(\text{b})=2\ln(\text{a})\\ 3\ln(\text{d})=4\ln(\text{c})\\ \text{a}-\text{c}=9 \end{cases}\space\space\space\Longleftrightarrow\space\space\space \begin{cases} \text{b}=\text{a}^2\\ \text{d}^3=\text{c}^4\\ \text{a}-\text{c}=9 \end{cases}\space\space\space\Longleftrightarrow\space\space\space \begin{cases} \text{b}=\left(9+\text{c}\right)^2\\ \text{d}^3=\text{c}^4\\ \text{a}=9+\text{c} \end{cases} $$