How to solve this Diophantine equation (involving natural logarithms)?

Question

The equation is $r = \ln{a} + b \ln{c}$ where $r \in \mathbb{R}$ is fixed and $a,b,c \in \mathbb{N}$. In other words, for arbitrary real r, how can one say whether a solution (in form above) exists or not.

Answer 1

This is somewhat an ill-posed problem, because unless $a = c = 1$, your real number will be irrational and thus there will probably be no way to present the number other than writing it in terms of natural logarithms and rational numbers in the first place, in which case you can use a fairly straightforward algorithm to see if the expression in terms of natural logarithms and rational numbers can be written in your given form.