# How to compare big numbers that are outcome of different functions.

How is the best way to compare big numbers? They are result of two functions with different asymptotic growth. For example:

Googleplex which is $10^{{10}^{100}}$ to $1000!$

-
You might look at math.stackexchange.com/questions/72646/… – Ross Millikan Jul 22 '12 at 1:07
Take logarithms. Use Stirling's formula. – Qiaochu Yuan Jul 22 '12 at 1:32
Apropos to Ross's comment, see this article as well. – J. M. Jul 22 '12 at 3:55

$10^{googol}$ compared to $1000!$
$1000!=1000\times999\times998...<1000^{1000}$
$1000^{1000}=(10^3)^{1000}=10^{3000}$
since a googol is drastically larger than $3000$, the first number is much, much greater.
In general logarithms (equivalently, converting to a base and comparing exponents) are a great way for comparing large numbers. For example: whether $2^{523} <^? 3^{228}$ may not be obvious, but even knowing very rounded values for $\log(2)$ and $\log(3)$ will let you compare $523\log(2)$ and $228\log(3)$ quite easily, which is an equivalent problem.