How Are They All Linked Together?

Algebra Level 2

$\large \begin{aligned} {4^{\color{#D61F06}a}=5} \\ {5^{\color{#3D99F6}b}=6} \\ {6^{\color{#20A900}c}=7} \\ {7^{\color{magenta}d} = 8} \\ \end{aligned}$

Supose $\color{#D61F06}{a}$ , $\color{#3D99F6}{b}$ , $\color{#20A900}{c}$ and $\color{magenta}{d}$ satisfy the system of equations above, find $\color{#D61F06}{a}\times \color{#3D99F6}{b}\times \color{#20A900}{c}\times \color{magenta}{d}$ .