# How does $n^c \lg n, 0<c<1$ compare to other common time complexities

Between what two common time complexities would you place $$n^c lg n, 0?

The following table illustrates the common time complexities. Source: wikipedia

$$O(n^c \log n)$$ is above $$O(n^c)$$ but below $$O(n^{c+\varepsilon})$$ for any $$\varepsilon > 0$$. This is true for any $$c \geq 0$$, including $$c < 1$$.
For example, $$n^{0.5} \log n$$ is not in $$O(n^{0.5})$$ but is in $$O(n^{0.500001})$$.
This is one of the reasons for the invention of the "soft O" notation $$\tilde{O}(n^c)$$, defined as the union of $$O(n^c (\log n)^d)$$ for all $$d \geq 0$$. Soft O allows you to fudge log factors and focus on the most important part, which is the highest-order polynomial exponent.