I must find the M.L.E for the Pareto distribution
$$f(x|x_0)=\begin{cases} \frac{\alpha x_0^{\alpha}}{x^{\alpha+1}} & \text{ if } x\geq x_0, \\ 0 & \text{ if } x<x_0. \end{cases}$$
I end up with a likelihood function
$$\alpha^n x_0^{n\alpha}\prod\limits^n_{i=1}{1\over x_i^{\alpha+1}}.$$
Using the natural logarithm, my answer goes goes exactly like this one.
However, my solution sheet for the excercise states that I can't use the logarithm on the likelihood function in this case because it is undefined. Instead, I must define $T=\min(X_1,\ldots,X_n)$ so the likelihood function ends up looking like this
$$\frac{\alpha^n T^{n\alpha}}{\prod_{i=1}^n x_i^{\alpha+1}}1_{\{T\geq x_0\}},$$
Where $1$ is the indicator function.
I have no idea what's going on. Every single solution I find for this excercise solves it as I did. Am I misinterpreting the problem or is the solution sheet is wrong? If so, why?