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Inside a previous question, one particular nested function shown is the known tetration. This "kind" of arbitrary repeated functions has always intrigued me, because inside their properties lie so many beautiful relations (i.e. the domain of $\lim_{n\to+\infty}{}^nx$ is $e^{-e}\leq x\leq e^{1/e}$).

So I wanted to define a similar expression, and search for its properties: $$ f(n,x)={}^n\log x=\underbrace{\log(\log(\log(\ldots\log x)))}_{n\text{ times}},\quad n\in\mathbb{N}^+,x\in\mathbb{R} %\quad\text{or}\quad{}^n\sin x=\underbrace{\sin(\sin(\sin(\ldots\sin x)))}_{n\text{ times}} $$ just looking at how the domain explodes even when $n<8$ baffles me, as I obtained: $$ \mathcal{D}_{f(n\geq2,x)}:x>{}^{n-2}e\wedge\mathcal{D}_{f(1,x)}:x>0\quad\text{for example, }\;\mathcal{D}_{f(7,x)}:x\gtrsim 10^{10^{10^{6.22}}} $$ This may be not the only "strange" property of this particular function, but even if I think it can be exploited, I've never found any source (neither textbooks nor online); so in conclusion, is there any application of such function (or functions' class)?

Community
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TheVal
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2 Answers2

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Recently, it was established by Kevin Ford, Ben Green, Sergei Konyagin, Terence Tao, and also by James Maynard, that there are prime gaps below $N$ as large as $$c\frac{\log N\log\log N\log\log\log\log N}{(\log\log\log N)^2}$$ when $N$ is large enough, and they showed that $c$ could be as large as you like.

Empy2
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Let us examine this.

$^{1}\log(x)$ only yields a real number for positive real $x$. For other $x$, they are never never real.

$^{2}\log(x)$ only yields a real value for real values of $x>1=10^0$ It is undefinied for $x=1$, and has only nonreal values otherwise.

$^{3}\log(x)=\log(^2\log(x))$ Taking the antilog of the inequality above, we can deduce that it only returns a real value for $x>10=10^1$

$^{4}\log(x)=\log(^3\log(x))$ Taking the antilog of the inequality above, we can deduce that it only returns a real value for $x>10^{10}=10^{10^1}$

Note that for each increase of $n$ by 1, the minimum value for $x$ for a real result is the common antilog of the minimum value the the previous.

$$^{n}\log(x)=log(^{n-1}log(x))$$ The minimum value of $x$ for this to be a real number is equal to taking the common antilog of 1 $n-2$ times.

Michael Ejercito
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