"Information function" help? Thanks

Here's the question

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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.

$\displaystyle P=\{(p_{1},p_{2},...,p_{N})1^N|\sum_p{i}=1, p{i}>0\}$

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

$\displaystyle H(p_{1},p_{2},...,p_{N})=-\sum_{p_{i}=1}^{N} p{i}\log{p{i}}$

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.

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Here's what I did

$\displaystyle

Let p{i}=\frac{1}{N}$

$\displaystyle -\sum_{p_{i}=1}^{N} p{i}\log{p{i}}=[(-\frac{1}{2}\log\frac{1}{2})+(-\frac{1}{3}\log\frac{1}{3})+...+(-\frac{1}{N}\log\frac{1}{N})]$

$\displaystyle =[(\log2^\frac{1}{2})+(\log3^\frac{1}{3})+...+(\log{ N}^\frac{1}{N})]

$

Then I stuck here.

Did I head to wrong direction?

How should I solve this?

Thank you.