If \(X \sim Mult\left(n, \pi\right)\) and we observe \(X = x\), then the loglikelihood function for \(\pi\) is

\(L(\pi)=\log\dfrac{n!}{n_1!\cdots n_k!}+x_1 \log\pi_1+\cdots+x_k \log\pi_k\) 

We usually ignore the leading factorial coefficient because it doesn't involve \(\pi\) and will not influence the point where \(L\) is maximized. Using multivariate calculus with the constraint that

\(\pi_1+\ldots+\pi_k=1\)

the maximum is achieved at the vector of sample proportions:

\begin{align} \hat{\pi} = \dfrac{1}{n}x= (x_1/n,x_2/n,\ldots,x_k/n)\\ \end{align}