You'd better keep a log of that

Let the ratio between consecutive terms of the geometric sequence be $d$ such that $a_1 = k$ , $a_2 = kd$ , $a_3 = kd^2$ etc.

$\displaystyle \sum_{n=1}^{2018} \log_k(a_n) = 4072324$ $\log_k(a_1) + \log_k(a_2) + \log_k(a_3) + ... + \log_k(a_{2018}) = 4072324$ $\log_k(k) + \log_k(kd) + \log_k(kd^2) + ... + \log_k(kd^{2017}) = 4072324$ Now using the addition/multiplication law for logs: $\log_k(k \times kd \times kd^2 \times ... \times kd^{2017}) = 4072324$ $\log_k(k^{2018} \times d^{\frac{2017 \times 2018}{2}}) = 4072324$ $\log_k(k^{2018} \times d^{2035153}) = 4072324$ Again using the addition/multiplication law for logs: $\log_k(k^{2018}) + \log_k(d^{2035153}) = 4072324$ $2018 + \log_k(d^{2035153}) = 4072324$ $\log_k(d^{2035153}) = 4070306$ $2035153\log_k(d) = 4070306$ $\log_k(d) = 2$ $d = k^2$

As $\dfrac{a_{2019}}{a_{2017}}$ is the ratio between terms which are two apart, it is $d^2$ , making it $\boxed{k^4}$ .