If $$a^2=b^3=c^4$$, $$b^2=d^5$$, then find the the value of $$\log_ab$$ $$\times\ \ $$ $$\log_cd$$.

We have, $$a^2=b^3=c^4$$, $$b^2=d^5$$
=> d=$$b^{\frac{2}{5}}$$
Also, c= $$b^{\frac{3}{4}}$$ and a = $$b^{\frac{3}{2}}$$
Now, $$\log_ab$$*$$\log_cd$$ = $$\frac{\log b}{\log a}\times\ \frac{\log d}{\log c}$$
= $$\dfrac{\log\ b}{\log b^{\frac{3}{2}}}\times\ \dfrac{\log b^{\frac{2}{5}}}{\log b^{\frac{3}{4}}}$$

= $$\frac{\log\ b}{\frac{3}{2}\log b}\times\ \frac{\frac{2}{5}\log b}{\frac{3}{4}\log b}$$ = $$\frac{2}{3}\times\ \frac{2\times\ 4}{3\times\ 5}=\ \frac{\ 16}{45}$$