TITLE: Without using a calculator and logarithm, which of $100^{101} , 101^{100}$ is greater?
QUESTION [17 upvotes]: Which of the following numbers is greater? Without using a calculator and logarithm.
$$100^{101} , 101^{100}$$
My try :  $$100=10^2\\101=(100+1)=(10^2+1)$$
So  :
$$100^{101}=10^{2(101)}\\101^{100}=(10^2+1)^{100}=10^{2(100)}+N$$
Now what ?

REPLY [6 votes]: In this answer, it is shown that $\left(1+\frac1n\right)^n$ increases to $e\lt3$. Thus, for $n\ge3$,
$$
\left(1+\frac1n\right)^n\lt n
$$
Multiply by $n^n$ and we get
$$
(n+1)^n\lt n^{n+1}
$$