Exponential Maxima

Let $X_1,X_2,\dots \sim \exponential{\lambda}$ IID. Let $p_n = \mathbb{P}\left( M_n \leq \dfrac{\ln(n)}{\lambda} \right)$, where $M_n = \max(X_1,\dots,X_n)$. Compute $\displaystyle \lim_{n\rightarrow \infty} p_n$ when $\lambda = 4$. The answer is in the form $e^{-a}$ for some $a$. Find $a$.