  Let $\{L(t),t\geq 0\}$ be a L\'{e}vy process with representative random
variable $L(1)$ defined by the infinitely divisible logarithmic series
distribution. We study here the transition probability and L\'{e}vy measure of
this process. We also define two subordinated processes. The first one, $Y(t)$,
is a Negative-Binomial process $X(t)$ directed by Gamma process. The second
process, $Z(t)$, is a Logarithmic L\'{e}vy process $L(t)$ directed by Poisson
process. For them, we prove that the Bernstein functions of the processes
$L(t)$ and $Y(t)$ contain the iterated logarithmic function. In addition, the
L\'{e}vy measure of the subordinated process $Z(t)$ is a shifted L\'{e}vy
measure of the Negative-Binomial process $X(t)$. We compare the properties of
these processes, knowing that the total masses of corresponding L\'{e}vy
measures are equal.
