what's the distribution of $x \log(x)$? If $x$ is a Binomial distributed random variable, what's the distribution of $x \log(x)$? P.S. if $x=0$, we take $x\log(x)=0$.
EDIT: what if we had $p \log(p)$, where $p$ is an estimated probability (not specifecally the binomial parameter)?
 A: When you transform a discrete random variable, all you do is move the spikes of probability around. If the probability at $X=x$ is $p(x)$ and you transform $Y=t(X)$, then at $y=t(x)$ you have probability $p(x)$. 
If the transformation is not injective then where more than value is mapped to a single transformed value the probabilities will all add.
Apart from the fact that both 0 and 1 map to 0, the transformation $Y=X\log(X)$ where $X$ can take the values $\{0,1,2,...,n\}$ is otherwise monotonic (it's monotonic in the positive integers; as a transformation of a continuous variable it would be monotonic for $x > 1/e$).
So you have
x   0     1    2     3    ...    n
p  p(0)  p(1) p(2)  p(3)  ...   p(n)
    \   /       |    |           |      
y     0      2log2 3log3  ...  nlogn
p  p(0)+p(1)  p(2)  p(3)  ...   p(n)

It's possible to write the probabilities in closed form for $y = 2\log 2$ and up but it's not really more enlightening than just working with probabilities for $x$ and then shifting the locations.
That is we can write $P(Y=0)= (1-p)^{n-1}(1 + (n-1) p)$
and $ P(Y=y) = {n \choose e^{W(y)}} p^{e^{W(y)}} (1-p)^{n-{W(y)}}$, $y=2\log 2,3\log 3, ...,n\log n$, where $W$ is the Lambert $W$-function -- but it's less clear than writing the cases after the first one as 
$P(Y=y) = {n \choose x} p^{x} (1-p)^{n-x}\,$, $x=2,3,...,n$ where $y=x\log x$.
A: If $X$ follows a discrete distribution, then you can think of its probability mass function as a table with two columns: one for $x_i$ values and the second one for $p_i$ probabilities assigned to the values. Now if you change the $x_i$ values in some way that maps them to other unique values, then it does not change anything about the $p_i$ probabilities. Think of it in terms of counts from a sample, if you counted eleven $x_i$ values in your sample, then if you re-tag them to $x_i \log x_i$, there are still eleven of them. The only exception is $0$, as noted by Bad John.
To convince yourself you can simulate draws from the binomial distribution and transform them using $x \log x$ function, and then plot the empirical probabilities against the corresponding probabilities $f(x)$ -- they will be the same, only the labels of the $x$-axis will change.

As about $p \log p$, as already noted in the comments, the question is not clear. If the question is that you know what is $p \log p$ and want to deduce binomial probability $p$ of it, then it cannot be done. As you can see from the plot below, for each $p \log p$ you have two corresponding $p$ values that can lead to it, so there is no inverse function.

A: First of all, p is a parameter. So, your second question is unclear.In case of trying to find the distribution of $Y = X\log X$, then you have to think that $X$ is the discrete variable $Bin(n,p)$. Then, $Y$ takes the values $x log x$ for $x=0,1,...,n$ with probabilities $P(X=x)$. The interesting point is when $x=0$, but if you take the $\lim_{x\to 0} \, x \log (x) = 0$, you can suppose that when $x=0$ then $y=0$. In my opinion, $Y$ is not a known distribution but it is clearly a discrete distribution with $P(Y=0) = P(X=0) + P(X=1)$ and  $P(Y=y) = P(X=x)$ for $x=1,2,...n$.
I made an R code to see the results. The red line is the Binomial distribution and the blue line is the $Y=X \log X$ distribution.
$Y=Xlog X$ distribution" />
Note that nevertheless I have joined the points, the distributions are discrete.
