Taking Log to find MAXIMIZE summation of variables I have been reading IEEE papers on communication and in several papers the authors formed objective function like: 
$\text{Maximize }    \sum_k \log r_k $
to maximize the total rate of the system of $k$ users. Here $r_k$ is the rate of the $k$'th user. Now I cant understand how taking logarithm is relevant with maximizing. This is a common case in 'Entropy Maximization'. Is there any general mathematical explanation of this? Any explanation is highly appreciated.
 A: Maximizing a sum of logs over a convex set is called achieving proportional fairness. This is a standard goal in communication networks.  Here, $r_k$ is the communication rate achieved by user $k$.  Each user wants to maximize its own rate.  However, there are shared resources.  Let $\Lambda$ be the (typically convex) set of all possible rate vectors $(r_1, r_2, \ldots, r_K)$ that the system can support.  We want to find a Pareto optimal point.  Maximizing a sum of increasing and concave utility functions is used because concave functions have a diminishing returns property, which says that we would prefer to make allocations relatively even (pushing up those who have lower rate). 
The log function is analyzed in particular in: 
Frank Kelly, "Charging and Rate Control for Elastic Traffic," European Transactions on Telecommunications, vol. 8, no. 1 pp. 33-37, Jan.-Feb. 1997. 
If $\Lambda$ is a convex set, it can be shown that finding a vector $r^*=(r_1^*, \ldots, r_K^*)$ that maximizes $\sum_{k=1}^K \log(r_k)$ subject to $(r_1, \ldots, r_K) \in \Lambda$ means that: 
$$ \sum_{k=1}^K\frac{r_k-r_k^*}{r_k^*} \leq 0 $$
for all vectors $(r_1, \ldots, r_K) \in \Lambda$.  This can be interpreted as the "proportionally fair" criterion:  Consider changing from the $r^*$ vector to some other vector $r \in \Lambda$. You might be able to increase rates of particular users by deviating from  $r^*$, but the overall proportional change is never positive, where this change weights users with currently low rates $r_k^*$ more heavily, which is a fairness feature. 

Example 1: A simple example is when the set $\Lambda$ is the set of all nonnegative rates that sum to less than or equal to 1.  Then one solution to maximizing the linear function $\sum_k {r_k}$ over $r \in \Lambda$ is $(1, 0, 0, ..., 0)$, which is clearly unfair since it gives one user everything.  Howevever, maximizing the strictly concave function $\sum_k \log(r_k)$ over the same set $\Lambda$ gives the (much more fair) solution $r^*=(1/K, 1/K, ..., 1/K)$.  
Example 2:  Suppose $\Lambda$ is the set of all $(r_1, ..., r_K) \geq 0$ such that $r_1/2 + \sum_{k=2}^K r_k \leq 1$.  Maximizing the linear utility function $\sum_k r_k$ over $\Lambda$ gives the (unfair, and unique)  solution $(2, 0, 0, ..., 0)$.  Yet, maximizing the $\sum_k \log(r_k)$ over the same set gives the (more fair, in fact, "proportionally fair") solution: $(\frac{2}{K},  \frac{1-1/K}{K-1}, ..., \frac{1-1/K}{K-1})$. 
