Which one is the larger , $10^{30!}$ or $10^{30}!$?

Question

So, I have got this problem in which one is asked to find the greater one between $10^{30!}$ or $10^{30}!$.

Taking $\log$ both sides leads me nowhere.

Answer 1

You'll want Stirling's approximation here: $\ln n! \approx n \ln n - n$. Here $\ln$ is the natural log.

So $\ln (10^{30})! \approx 10^{30} \ln 10^{30} - 10^{30}$. Now $\ln 10^{30} = 30 \ln 10 < 30 \times 3 < 100$, and so $\ln (10^{30})! < 10^{32}$.

On the other hand, $\ln 10^{30!} = 30! \ln 10$. You can verify that $30! > 10^{32}$. and so $\ln 10^{30!} > 10^{32}$.

Thus $10^{30!}$ is the larger of the two.

Answer 2

$x! < x^x\\ 10^{30}! < (10^{30})^{10^{30}}$

While $10^{30!} = (10^{30})^{29!}$

Since $10^{30} > 1$, if $29! > 10^{30}$ (which it is) then $10^{30!} > 10^{30}!$