"Information function" help? Thanks

Here's the question

---------------------------------

Let I be the unit interval, i.e. I = {x | 0 x 1}, and let I N be the N-fold cross product of I with itself, i.e., the unit N-box. Let P be that subset of I N consisting of points with positive coordinates summing to 1, i.e.

Let R be the real numbers and define the information function H : P R by

H gives the amount of information in a communication system with N alternative messages where the ith message is transmitted with probability pi. (Interesting fact: If the base of the logarithm is chosen to be 2, then the unit of information is the bit, and corresponds to the amount of information in one yes-no question.) Prove that if all the messages are transmitted with equal probability, then the amount of information is equal to log N.

--------------

Here's what I did

Then I stuck here.

Did I head to wrong direction?

How should I solve this?

Thank you.