How Are They All Linked Together?

I thought I can post a method without using logarithms , which may not appear simpler than the remaining solutions.

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

$\large\begin{aligned} \therefore\color{#D61F06}{4^{a\color{#3D99F6}b\color{#20A900}c\color{magenta}d}}&=8=4^{3/2}\\ \implies\color{#D61F06}a\times\color{#3D99F6}b\times\color{#20A900}c\times\color{magenta}d &=\dfrac 32 \end{aligned}$

$x^y=z\implies y=\log_x z=\dfrac{\ln z}{\ln x}$ $\Large{\color{#D61F06}{4^a=5} \implies a=\dfrac{\ln 5}{\ln 4}\\ \color{#3D99F6}{5^b=6} \implies b=\dfrac{\ln 6}{\ln 5}\\ \color{#20A900}{6^c=7}\implies c=\dfrac{\ln 7}{\ln 6}\\ \color{magenta}{7^d = 8}\implies d=\dfrac{\ln 8}{\ln 7}}$

$\color{#D61F06}{a}\times \color{#3D99F6}{b}\times \color{#20A900}{c}\times \color{magenta}{d}=\dfrac{\cancel{\ln 5}}{\ln 4}\times\dfrac{\cancel{\ln 6}}{\cancel{\ln 5}}\times \dfrac{\cancel{\ln 7}}{\cancel{\ln 6}}\times\dfrac{\ln 8}{\cancel{\ln 7}}$ $\large=\dfrac{\ln8}{\ln4}=\dfrac{3\cancel{\ln 2}}{2\cancel{\ln2}}=\huge\boxed{\dfrac{3}{2}}$

$a \times b \times c \times d = \log_{4} 5 \times \log_{5} 6 \times \log_{6} 7 \times \log _{7} 8 = \log_{4} 8 =$ $\log_{2^2} 2^3 = \frac{3}{2}$