Set $n = p_1^{e_1} \cdot p_2^{e_2} \cdot ... \cdot p_k^{e_k}$. Since $p_1, p_2, ... \geq 2$ and $e_1, e_2, .. \geq 1$, $n \geq 2^1 \cdot 2^1 \cdot ... = 2^k$. Since ln is an increasing function, $\ln n \geq \ln 2^k = k \ln{2}$.