Prime numbers/Logarithms/Linearly independent/Tip/Exercise

We consider the real numbers ${}\mathbb {R}$ as a ${}\mathbb {Q}$-vector space. Show that the set of real numbers ${}\ln p$, where ${}p$ runs through the set of all prime numbers, is linearly independent. Tip: Use that every positive natural number has a unique representation as a product of prime numbers.

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