How to show that $n^{\ln(\ln(n))} = \ln(n)^{\ln(n)}$ I have verified that $n^{\ln(\ln(n))} = \ln(n)^{\ln(n)}$ by plugging in values for $n$, but do not understand why it is true. I am not aware of any $\log$ rules that can be used to simplify $n^{\ln(\ln(n))}$.
Note that $n=e^{\ln n}$ so

$$n^{\ln (\ln n)}=e^{\ln n\cdot \ln(\ln n)}=(\ln n)^{\ln n}$$
Following Cameron Williams' hint, we take the logarithm of both sides to get

$$\ln \left(n^{\ln\left(\ln (n)\right)}\right) = \ln \left( \ln (n)^{\ln (n)}\right)$$

Recall that $\log_b a^c = c\log_b a$. Use this property on both sides:

$$\left(\ln\left(\ln(n)\right)\right)\left(\ln(n)\right)=\left(\ln(n)\right)\left(\ln\left(\ln(n)\right)\right)$$

This is true because multiplication is commutative.