# Domain of the n composed logarithms on x.

As we know, the real logarithm has the domain

$$D_1 = \{x : x \in \mathbb{R}, x > 0\}$$

What is the logarithmic domain of "higher order" logarithms, at index n? For example, it seems that

$$D_2 = \{x : x \in \mathbb{R}, x > 1\}$$

which would be the domain of

$$f(x) = \log(\log(x))$$

$D_3$ is a little hard to imagine. I think my real question deals with formalization, and the extended case of $D_n$ however.

There may be connections to the iterated logarithm, which says how many iterations are necessary before a value breaks in a certain base. See the wikipedia article.

• Yes, there is a connection. Which base did you have in mind? Nov 14 '15 at 23:08
• base e? I was kind of asking abstractly, if you have an answer in an arbitrary base that would be impressive and helpful. Nov 14 '15 at 23:08
• Hint: If $\ln^{(n)}(x)$ denotes the iterated logarithm and ${^n}e=e^{e^{\ldots{e}}}$ ($n$-times) denotes tetration, then the following identity holds for all $n\in\mathbb{N}$: $\ln^{(n)}({^n}e)=1$. Nov 14 '15 at 23:34
• @YiannisGalidakis, do you mean the super logarithm (inverse of tetration)? Nov 14 '15 at 23:37
• I think so, yes, although I am not very familiar with the actual definition Andrew Robbins uses. It's bullet 3 on that page :As the number of times a logarithm must be iterated to get to 1 (the Iterated logarithm) @theREALyumdub Nov 14 '15 at 23:42

If you denote by $\ln^{(n)}(x)$ the iterated logarithm and by ${^n}e=e^{e^{\ldots^{e}}}$ (height $n$) iterated exponentiation of the base $e$ (as per the comment), we have by definition:

$$\ln^{(n)}({^n}e)=1\Rightarrow$$ $$\ln^{(n+1)}({^n}e)=\ln(1)=0$$

Apply for $n=1$ and we get:

$$\ln(\ln(e))=0$$

So $D_2$ should be $D_2=\{x\in\mathbb{R}\colon x\ge e\}$. This however breaks the pattern, because the range of $\ln(\ln(x))$ can be negative for this case, if we extend the domain of $x$ to be $x\gt 1$.

You can't do this for higher iterates however, because negative ranges are not allowed in the domain of $\ln$. Therefore:

$$D_1=\{x\in\mathbb{R}\colon x\gt 0\}$$ $$D_2=\{x\in\mathbb{R}\colon x\gt 1\}$$ $$D_3=\{x\in\mathbb{R}\colon x\ge {^2}e\}$$ $$D_4=\{x\in\mathbb{R}\colon x\ge {^3}e\}$$

and in general ($n\ge 3$):

$$D_n=\{x\in\mathbb{R}\colon x\ge {^{n-1}}e\}$$

• Much more formal than my answer; I was so busy trying to edit mine to be correct that I missed this. Nov 15 '15 at 1:35

Actually, now that I've looked into it, the table on the page seems to suggest that the iterated logarithm values are the way you compute the bounds on the $D_n$ domains, and it switches at the tetration integer values.

## The tetration operation on the base defines the bounds of the domains $D_n$.

Let b be the base of the logarithm in question, so that $f_1(x) = \log_b(x)$

We can say

$$D_1 = \{x : x \in \mathbb{R}, x > 0\}$$ $$D_2 = \{x : x \in \mathbb{R}, x > b\}$$

And for n > 2, we apply the iterated logarithm to say that

$$D_3 = \{x : x \in \mathbb{R}, x > b^b\}$$ $$D_4 = \{x : x \in \mathbb{R}, x > (b^b)^b\}$$ $$...$$ $$D_n = \{x : x \in \mathbb{R}, x > tetra(b,n - 1)\}$$

• Yes, that looks ok. Nov 15 '15 at 0:03
• The exponent should probably be $n-1$. Your $D_2$ is wrong, since $\ln(\ln(1))$ has problems. It should be $x>e$ for base $e$. Then $D_n=\{x:x\in\mathbb{R},x>tetra(e,n-1)\}$ for base $e$. Nov 15 '15 at 0:14
• Does that look better? Nov 15 '15 at 1:32
• Your $D_2$ needs correction. It is $x>1$ (it is the only one that breaks the pattern, see my answer. My comment above has a typo) and when you notate tetration it's prettier to stick to one of the two forms you have: $tetra(b,3)=b^{b^{b}}$, but parentheses go from top to bottom. So it'd be $b^{(b^b)}$, $b^{(b^{(b^b)})}=tetra(b,4)$, etc. Nov 15 '15 at 1:55