Comparing $(2n)^{\ln {n}}$ to $(\ln{n})^{2n}$ for $n > 1000$

Question

I want to compare these two numbers and prove which is bigger for $n > 1000$:

first: $(2n)^{\ln {n}}$

second: $(\ln{n})^{2n}$

I tried to somehow simplify them to similiar form and do induction, but I wasn't successful.

Answer 1

In a very naive way, the first one has a logarithm to the power. The second one has an exponential behaviour, though having a logarithmic base. THe second one will of course run faster (very faster) than the first one, hence for large values of $n$ we have for sure:

$$(\ln(n))^{2n} > (2n)^{\ln(n)}$$

A more thought approach is the following.

Consider: take the logarithm of both quantities:

$$\ln\left((2n)^{\ln(n)}\right) = \ln(n)\ln(2n) \sim \ln^2(n)$$

$$\ln\left(\ln(n)^{2n}\right) = 2n\ln(\ln(n))$$

Now since we are dealing with $n > 1000$ we can think of $n = e^7$ whence:

$$\ln^2(n) \longrightarrow \ln^2(e^7) = 49$$

$$2n \ln(\ln(n)) \longrightarrow 2e^7 \ln(\ln(e^7)) = 2e^7\ln(7)$$

The second one is surely greater because of the linear part $2n$ that here takes the whole $e^7$ into account.