# $\{\ln(p)\}$ where $p$ is prime is linear independent. [duplicate]

I’m struggling to prove that the set $\{\ln(p): \text{$p$is prime}\}$ is a linearly independent system in the Rational vector space of the real numbers.

Any help is greatly appreciated.

## marked as duplicate by Stefan Mesken, Martin Sleziak, Crostul, Siong Thye Goh, David KDec 11 '17 at 19:21

Hint: Use contradiction. suppose it is not linearly independent. $$\ln(p_i) = \alpha_1\ln(p_j) + \alpha_2\ln(p_k) = \ln(p_j^{\alpha_1}p_k^{\alpha_2 })$$
So, $p_i = p_j^{\alpha_1}p_k^{\alpha_2 }$. As all $p_i$, $p_j$, and $p_k$ are prime, it can not be possible. To be more precise, you should use induction to show this.