I am trying to understand the space and time Hierarchy theorems according to
Sanjeev Arora, Boaz Barak: Computational Complexity: A Modern Approach
but the more general case.
What I don't understand is why the language decided by our custom machine is in the larger complexity class. For example, with the case of time complexity, the cost of simulating the input machine is $t_1 \log t_1$, so why is the language decided by it in $DTIME(t_2)$ but not in $DTIME(t_1 \log t_1)$?