Master theorem: $T(n)=10T(n/9)+n\lg(n)$
Case 3 does not apply. Indeed:
$$
f(n) = n \log n \not\in \Omega(n^{\log_9 10}) = \Omega(n^{\log_b a}).
$$
However case $1$ applies since, for $0&lt;c \le 0.01$
$$
f(n) = n \log n \in O(n^{1.04 - 0.01}) \subset O(n^{\log_9 10 - 0.01} ) \subseteq O(n^{\log_b a - c} ).
$$
This shows that $T(n) \in \Theta(n^{\log_9 10})$.