How can I solve for $n$ in the equation $n \log n = C$?

If what you want is to solve for $n$, there is no simple way. The solution has $n=e^{W(C)} = \frac C{W(C)}$ where $W()$ is the Lambert W function.

Since $n \ln n = \ln n^n$, just raise $e$ to each side and you get $n^n = e^C$.

If you want to solve for $n$, I don't know of any method that will work (usually in Computer Science we use approximations or we solve it numerically).

I will take the base of the logarithm to be $e$. (If you are in a different base; just replace the $e$.)

$$n\log n = C$$

$$\log n = \frac{C}{n}$$

$$e^{\frac{C}{n}} = n$$

$$(e^C)^{\frac{1}{n}} = n$$

$$e^C = n^n$$