theorem, efficient universal Turing machine

Theorem

There exists a TM $\mathcal{U}$ such that for every $x, \alpha \in \{0,1\}^*$, $\mathcal{U}(x,\alpha) = M_\alpha(x)$, where $M_\alpha$ denotes the TM represented by $\alpha$.
Moreover, if $M_\alpha$ halts on input $x$ within $T$ steps then $\mathcal{U}(x,\alpha)$ halts within $CT\log T$ steps, where $C$ is a number independent of $\lvert x \rvert$ and depending only on $M_\alpha$'s alphabet size, and number of states.

References

1. S. Arora, B. Barak. Computational Complexity: A Modern Approach, Cambridge University Press, 2009, p. 20.
2. https://www.cs.princeton.edu/courses/archive/spr06/cos522/lec2.pdf