From Baker's theorem it follows that a linear combination of natural logarithms of prime numbers with non-zero algebraic coefficients can never be zero.
Has it been proved that the set of all natural logarithms of prime numbers is algebraically independent over $\mathbb Q$?
In other words, can we be sure that expressions like the following are never exactly zero? $$\frac{347}{75}\,\ln^22\cdot\ln3-\frac{173}{100}\,\ln2\cdot\ln^23+\frac{179}{180}\,\ln^32-\ln^33$$