I have two unknown distributions
\begin{align*} f(x;\theta) &= \frac{1}{x\cdot \ln \theta} & 1 < x < \theta \end{align*} and
\begin{align*} f(x;\beta) = \frac{\beta}{(1+x)^{\beta+1}} \hspace{2cm} x > 0 \end{align*}
And I am trying to find the sufficient statistic for x as follows; Find the sufficient statistic $u(X_1,X_2,...,X_n)$ for $\beta$ and $\delta$.
So far I have tried to simplify it to follow the $\phi$ and $h$ format but I am stuck.
I have gotten as follows for the first distribution
\begin{equation*} f(x_1,x_2,x_3,...,x_n;\theta) = f(x_1;\theta) \cdot f(x_2;\theta) \cdot f(x_3;\theta)\cdot ... \cdot f(x_n;\theta) \end{equation*} \begin{equation*} f(x_1,x_2,x_3,...,x_n;\theta) = \frac{1}{(\ln \theta)^n \cdot \prod_{i=1}^n x_i} = \frac{1}{(\ln \theta)^n} \cdot \frac{1}{\prod_{i=1}^n x_i} \end{equation*}
and here for the second.
\begin{equation*} f(x_1,x_2,x_3,...,x_n;\beta) = \prod_{i=1}^n\Big(\frac{\beta}{(1+x_i)^{\beta+1}}\Big) \end{equation*}
\begin{equation*} f(x_1,x_2,x_3...x_n;\beta) = \Big(\frac{\beta^n}{\prod_{i=1}^n (1+x_i)^{\beta + 1}}\Big) \end{equation*}
Am I missing any steps or is there an easier way to approach this problem?
Factorization theorem says that if joint density $f_{\theta}$ of $(X_1,\ldots,X_n)$ factors as
$$f_{\theta}(x_1,\ldots,x_n)=g(\theta,t)h(x_1,\ldots,x_n)\,,$$
where $g$ depends on $\theta$ and on $x_1,\ldots,x_n$, where the dependence on $x_1,\ldots,x_n$ is through $t=t(x_1,\ldots,x_n)$ only, and $h$ is independent of $\theta$, then $t(X_1,\ldots,X_n)$ is a sufficient statistic for $\theta$.
It is important to keep track of the support of the joint distribution, especially when the support depends on the unknown parameter.
So for the first problem, joint density of $X_1,\ldots,X_n$ (assuming independence) is
\begin{align} f_{\theta}(x_1,\ldots,x_n)&=\prod_{i=1}^n \frac1{x_i\ln\theta}\mathbf1_{1<x_i<\theta} \\&=\frac{1}{(\ln\theta)^n\prod_{i=1}^n x_i}\cdot \mathbf1_{1<x_1,\ldots,x_n<\theta} \\&=\underbrace{\frac{\mathbf1_{1<x_{(n)}<\theta}}{(\ln\theta)^n}}_{g(\theta,t)} \cdot \underbrace{\left(\prod_{i=1}^n x_i\right)^{-1}}_{h(x_1,\ldots,x_n)} \end{align}
Here $x_{(n)}=\max\{x_1,\ldots,x_n\}$ and $\mathbf1_{x\in A}$ is an indicator function which equals $1$ if $x\in A$ and equals $0$ if $x\notin A$.
You can see $g(\theta,t)$ depends on $\theta$ and it depends on $x_1,\ldots,x_n$ through $t=x_{(n)}$.
Do this for the second problem. If you have any difficulty in the algebra, simply solve the problems for $n=2$ first to see the pattern.