smc

We see that we can re-write $4^{x_1}=5$, $5^{x_2}=6$, $6^{x_3}=7$, ... , $127^{x_{124}}=128$ as ${\left(...\left({\left({\left(4^{x_1}\right)}^{x_2}\right)}^{x_3}\right)...\right)}^{x_{124}}=128$ by using substitution. By using the properties of exponents, we know that $4^{x_1{x}_2...x_{124}}=128$. $$\begin{gathered} 4^{x_1{x}_2...x_{124}}=128 \\ {2}^{2x_1{x}_2...x_{124}}=2^7 \\ 2x_1{x}_2...x_{124}=7 \\ {x}_1{x}_2...x_{124}=\frac{7}{2} \end{gathered}$$ Therefore, the answer is $\boxed{\frac{7}{2}}$