1. Let $t=\log _{x^{2019}} 2018$, then the equation will take the form: $t+t^{2}+t^{3}+\ldots=2018$.

Simplify the left part of the last equation using the formula for the sum of an infinite decreasing geometric progression, we get: $\frac{t}{1-t}=2018$.

From which $t=\frac{2018}{2019}$.

We get, $\log _{x^{2019}} 2018=\frac{2018}{2019},\left(x^{2019}\right)^{\frac{2018}{2019}}=2018, x=\sqrt[2018]{2018}$.
Answer: $\sqrt[2018]{2018}$.