# Is there a name for a function that "grows exponentially" on a log-scale?

The ordinate of the function below is plotted in log-scale. In this presentation, the function looks like it "grows exponentially". Is there a standard name to describe this growth?

[the values in the figure are: $$(1, 2, 3, 4, 5); (1^{1^1}, 2^{2^2}, 3^{3^3}, 4^{4^4}, 5^{5^5})$$]

• The process of taking “power towers” is called tetration. It is one of the hyper-operations Aug 2 at 18:55
• @Fshrike: The OP's sequence is not tetration growth, as that would be (in the case of base $2)$ ${}^{2}2=2^2,$ ${}^{3}2=2^{2^2},$ ${}^{4}2=2^{2^{2^2}},$ ${}^{5}2=2^{2^{2^{2^2}}}, \ldots$ (i.e. the tetrated exponent increases arithmetically). The OP's sequence is ${}^{3}2,$ ${}^{3}3,$ ${}^{3}4,$ ${}^{3}5, \ldots,$ and calling this tetration growth is analogous to calling $2^3,$ $3^3,$ $4^3,$ $5^3, \ldots$ exponential growth. (continued) Aug 2 at 20:37
• I'm pretty sure there is not a standard name for the OP's growth rate, but if I were trying to name it, I think it would be more useful to name the growth rate of $k^{m^1},$ $k^{m^2},$ $k^{m^3},$ $k^{m^4}, \ldots$ where $k$ and $m$ are fixed — maybe something like $(k,m)$-hyper exponential growth, with $(k,m,n)$-hyper exponential, etc. growths being defined as expected. The idea is that only the 3rd exponent level increases arithmetically, and not both the base and the exponents. Googling, I found this question which might be of interest. Aug 2 at 20:37
• Functions like $a^{b^x}$ are called doubly-exponential, and these appear exponential on a log-scale. Note that your function $f(x) = x^{x^x}$ is even faster than doubly-exponential; not sure about the word for this. Aug 2 at 21:06