How Are They All Linked Together?

Algebra Level 2

\[\large \begin{eqnarray} {4^{\color{red}a}=5} \\ {5^{\color{blue}b}=6} \\ {6^{\color{green}c}=7} \\ {7^{\color{magenta}d} = 8} \\ \end{eqnarray}\]

Supose \(\color{red}{a}\), \(\color{blue}{b}\), \( \color{green}{c}\) and \(\color{magenta}{d}\) satisfy the system of equations above, find \(\color{red}{a}\times \color{blue}{b}\times \color{green}{c}\times \color{magenta}{d}\).