I have a summation that involve log. I don't know how to solve this summation. I want to find an expression (even a good approximation is enough) for this summation.
$\sum_{k=0}^{n}{log(a_k)}$ or $log(\prod_{k=0}^{n}{a_k})$
I only know that $\sum_{k=0}^{n}{a_k}= N$
Any help Please?
Edit
Let $a_k$ is a random variable which can take its value according to binomial (or normal) distribution. Then how to solve the above log summation problem.
When you characterize this as a probability question rather than a general series question, if you avoid certain pitfalls you can use the Law of Large Numbers to characterize the sum. That is, having defined a probability distribution for $a_k$, this implies a probability distribution of $\log(a_k)$; then from the expectation and variance of $\log(a_k)$ you can estimate the expectation and variance of the sum for large $n$.
There is a discussion of the case where $a_k$ is (approximately) normally distributed on stats.SE (see this question) and a discussion of the case where $a_k$ has a binomial distribution on mathoverflow (see this question). One technique that is recommended is to use the Taylor series of $\log(x)$ centered at $\mathbb E(a_k)$.
You cannot actually let $a_k$ have a normal distribution, because any normal distribution has negative values. In fact, you really want to prevent $a_k$ from getting close to zero, because the resulting large negative logarithms will cause problems. (This is one of the above-mentioned "pitfalls" to avoid.) A binomial distribution with only positive integer values would produce a much better-behaved distribution of the logarithm.