Exponent notation (Tetration) What is the meaning of this notation, and how does it work?
$$\exp_{10}^2(1.09902),\,\exp_{10}^3(1.09902)$$
I knew this notation when i was reading about tetration on wikipedia. Here is the link:
Tetration
and the values ​​above are equal to the following values:
$$^4 3=\exp_{10}^2(1.09902)$$
$$^5 3=\exp_{10}^3(1.09902)$$
 A: As I read the article, $^43=\exp^3_{10}(1.09902)$.  You are off by $1$ in the exponent on the exponential function.  I am able to reproduce the number of digits of $^43$ using Alpha.  I am also able to get $10^{10^{1.09902}}\approx 3.638 \cdot 10^{12}$, so $^43\approx \exp^3_{10}(1.09902)=10^{10^{10^{1.09902}}}$.  I would now claim that the table in Wikipedia is technically wrong for $^53$ because $^53=3^{(^43)}$ while $\exp^4_{10}(1.09902)=10^{(^43)}$.  I haven't looked for what description of equality they are working with.  If you only care about orders of orders of magnitude these are very close.  The first thing that matters is the height of the stack, then next thing that matters is the top number, and the lower numbers matter less and less.  Still $3 \neq 10$, even if it is on the bottom.  I guess they would sweep that under the "values containing a decimal point are approximate."  When you are working with numbers this large you have to say what precision you are working to.
A: I think you have $$\Large{^43=\underbrace{3^{3^{3^3}}}_4=\exp_{10}^3(1.09902)={\underbrace{10^{10^{10}}}_3}^{\small{^{1.09902}}}}$$
