I have an optimization like below:

$\text{ minimize } \sum_k - \log_2 x_k $

$\text{subject to: } x_k \leq q , k =1,2, \cdots, N .$

I can form the Lagrange of the problem as below:

$L(x, \lambda) = \sum_{k=1}^{N} - \log_2 x_k + \sum_{k=1}^{N} \lambda_k (x_k -q) $

But How can I find KKT conditions for this? Moreover, How can I solve the problem analytically from there? I know I can solve it through some nonlinear optimization solver software but for now I am only interested in solving the problem analytically so that I can get an exact form of x.