I am trying to calculate the MLE for the logarithmic distribution. It holds
$$ P(X=k) = -\frac{p^k}{k \cdot \ln(1-p} $$
Thus, the ML function is
$$L_p(X_1,\dots, X_n) = \prod_{i=1}^{n} -\frac{p^{x_i}}{x_i \cdot \ln(1-p)} = p^{\sum_{i=1}^n x_i} \left(\frac{1}{\ln(1-p)}\right)^n \prod_{i=1}^n -\frac{1}{x_i} $$
and the log likelihood function is
\begin{align} \log L_p(X_1,\dots,X_n)& = \sum_{i=1}^n x_i \cdot \log(p) + n \cdot \log\left(\frac{1}{\log(1-p)}\right) + \log\left(\prod_{i=1}^n -\frac{1}{x_i}\right) \\&= \sum_{i=1}^n x_i \cdot \log(p) + n \cdot -\log(\log(1-p)) + \log\left(\prod_{i=1}^n -\frac{1}{x_i}\right) \end{align}
So:
$$\frac{\partial L_p}{\partial p} = \frac{1}{p} \sum_{i=1}^n x_i + n \frac{1}{\log(1-p)(1-p)} \overset{!}{=} 0$$
This is equivalent to:
$$\frac{1}{n} \sum_{i=1}^n x_i = - \frac{p}{\log(1-p)(1-p)}$$
Now I don't know how to go on. How do I get the estimator for $p$?
Thanks in advance, for helping !
Technically, I suspect that you have to write the log likelihood function as
$$ \sum_{i=1}^n x_i \cdot \log(p) + n \cdot \log\left(\frac{-1}{\log(1-p)}\right) + \log\left(\prod_{i=1}^n \frac{1}{x_i}\right) $$
since $log(1-p)$ is negative, so $log(\frac{1}{log(1-p)})$ is undefined. I got the same final equation as you for the MLE, though.
I don't see a way to get a closed form solution, so I think you need to use numerical methods/root-finding algorithms to find the root(s) of $$\frac{1}{\hat{p}} \sum_{i=1}^n x_i + n \frac{1}{\log(1-\hat{p})(1-\hat{p})} = 0$$