Ackerman numbers, arrow notation

Question

How to compare $3^{3^{3^3}}$ and to $3\uparrow (3\uparrow \uparrow 3)$. (Ackerman number, arrow notation)? Are these two numbers equal?And also, how to compare $3\uparrow (3\uparrow \uparrow 3)$ with googol and googolplex? Thank you.

Answer 1

Your first two numbers are equal. By definition $3\uparrow\uparrow 3=3^{3^3}$ and $3\uparrow n=3^n$, so $$\large3\uparrow(3\uparrow\uparrow 3)=3\uparrow 3^{3^3}=3^{3^{3^3}}\;.$$

Now $3\uparrow\uparrow 3=7,625,597,484,987$, so

$$\log_{10}\big(3\uparrow(3\uparrow\uparrow 3)\big)=7,625,597,484,987\log_{10}3>\big(7\times10^{12}\big)\log_{10}3>3\times10^{12}\;,$$

so

$$3\large\uparrow(3\uparrow\uparrow 3)>10^{3\times10^{12}}>10^{100}=1\text{ googol}\;.$$

On the other hand,

$$\log_{10}\big(3\uparrow(3\uparrow\uparrow 3)\big)=7,625,597,484,987\log_{10}3<10^{13}\;,$$

so

$$3\large\uparrow(3\uparrow\uparrow 3)<10^{10^{13}}<10^{10^{100}}=1\text{ googolplex}\;.$$