The Absence of a Multiplicative Logarithmic Property for Natural Logarithms

We investigate the multiplicative structure of natural logarithms, specifically of the form ln(a) · ln(b) = ln(c). The value c always takes the form of either a^ln(b) or b^ln(a), which are equivalent to each other in this context. Through the Four Exponentials Conjecture we are able to establish a conditional result that states for any fixed value of either a or b, the other will have at most one value where c will be non-transcendental. Due to the nature of transcendental numbers, this conditionally establishes that there is no polynomial equation consisting of a and b that can be used to reach c in every other case.