Inverse problem for a monoid with an exponential sequence of primes

In this paper, for an arbitrary monoid ${M(PE)}$ with an exponential sequence of primes $PE$ of type $q$, the inverse problem is solved, that is, finding the asymptotic for the distribution function of elements of the monoid ${M(PE)}$, based on the asymptotic distribution of primes of the sequence of primes $PE$ of type $q$.
To solve this problem, we introduce the concept of an arbitrary exponential sequence of natural numbers of the type $q$ and consider the monoid generated by this sequence. Using two homomorphisms of such monoids, the density distribution problem is reduced to the additive Ingham problem.
It is shown that the concept of power density does not work for this class of monoids. A new concept of $C$ logarithmic $\theta$-power density is introduced.
It is shown that any monoid ${M(PE)}$ for an arbitrary exponential sequence of primes $PE$ of type $q$ has $C$ logarithmic $\theta$-power density with $C=\pi\sqrt{\frac{2}{3\ln q}}$ and $\theta=\frac{1}{2}$.