Iteration of logarithm

Question

Searching "Iterated logarithm" seems to give results only about the function $\log*(x)$. I would like to read about the properties of functions such as $$\begin{align} \log_{1}(x)&=\log(x) \\ \log_{2}(x)&=\log(\log(x)) \\ \log_{3}(x)&=\log(\log(\log(x))) \\ \log_{-1}(x)&=\exp(x)\\ \log_{0}(x)&=x \end{align}$$ etc. Are there any resources available on such functions? For example, their target series or limit definitions.

Thanks,

A more standard notation for what you've called $\log_n(x)$ is $\log^{(n)}(x)$. In any case if $\log_n(x)$ were to converge to a limit $z$, then $z$ would be a fixed point of the function $\log(x)$. There are no such fixed points in the real domain as a quick graphical analysis should convince you. — hardmath, Sep 24 '21 at 21:11
Of possible interest is Robert Bryant, Note on the convergency of series, Proceedings of the London Mathematical Society 27 #539 (12 December 1895), pp. 69-70 (google-books link). The paper involves repeated forward and backward applications of the Cauchy condensation test for series. — Dave L. Renfro, Sep 24 '21 at 21:33
Also of possible interest: Convergence of series involving in iterated logarithms $\sum \frac{1}{n(\log n)^{\alpha_1}\cdots (\log^k(n))^{\alpha_k} }$ and Example where Ermakoff's convergence test is inconclusive and Series involving iterated logarithms by J. Marshall Ash (2009). — Dave L. Renfro, Sep 24 '21 at 21:59
If you try with $\log(1+x)$ instead you can construct iterations, and even fractional iterations, easily by considering the taylorseries for this function and its iterates. After gotten fluent with this, a modification introducing recentered powerseries around the fixpoint is then likely a good idea. — Gottfried Helms, Jul 11 '22 at 09:32

Sayan Dutta · Accepted Answer · 2021-09-26T07:00:21.640

You can't go on iterating the $\log$ function like you mentioned in your question. This is because the $\log$ function is not defined for negative numbers$^\#$, and it decreases so fast that even if you seed your iteration with a reasonably large value of $x$, very soon, you'll get a negative number and run into illegal domain.

So, to make sure, your iterations are well defined, we introduce the $\log^*$ function which has the same inspiration as yours, but keeps my point in mind. That's why, you increase by $1$ every time you take $\log$ and iteratively define $$\log^*(x)=1+\log^*(\log(x))$$

$\#$ Logarithm of negative numbers may be handled using complex numbers. But, I don't know how to do that. Also, what I saud about the $\log^*$ function is the standard inspiration to define such a function. Maybe, you don't get much by including complex numbers, or else somebody would have done that by now :)

But we can define the logarithm for negative numbers using complex analysis. Is this still not good enough? — Michael Riberdy, Sep 25 '21 at 22:42
@MichaelRiberdy oh you want to include complex numbers in this as well? Then I'm sorry, I can't help. I don't know complex logarithms. I should add that to my answer. — Sayan Dutta, Sep 26 '21 at 06:57

Gottfried Helms · Answer 2 · 2023-06-22T20:45:11.527

The following pictures might be interesting. I've done them some years ago; they show exemplaric iteration of the logarithm in the complex plane - however I used base $b=4$ and not $b=e$ here. But the logic of the iteration of the natural log should come out in the tendencies correctly.
For the fractional iteration I used the Schroeder-method for interpolation (which is explained elsewhere - here I just want to give an illustration for the intuition)

The first image takes as iteration-"root" firstly the complex line on the (imaginary) y-axis $(0,0î)...(0,0.5î)$ in $20$ steps of $0.025î$. Then these 20 coordinates were exponentiated which brought them towards right, roughly above the $(1,0î)$-coordinate. Then in that roughly "square" the $1/20$-height iterations are computed; connecting the points vertically and horizontall to lines give then the first grid, say "root-grid", with red-colored (Excel-) interpolated lines. All points from the root-grid are now taken to exponentiation (by base 4) giving the next grid to the right (grey lines).

The iterated logarithm to base 4 is done as well. Here the root-grid was logarithmized once: the occuring grid is colured green. Note: the origin $(0,0î)$ is not representable; but complex values around this point can anyway be logarithmized - they appear far out near the negative real axis.
After that again logarithmizing gives the blue grid and so on, moving (and contracting) towards the (complex) fixpoint. See this picture:

Picture 1:

A perhaps more intuitive picture is the following; I only focused on the iteration of the basic line $(0,0î)..(1,0î)$ (blue dotted line on the real axis). Logarithmizing must take into account, that $(0,0î)$ would map to negative infinity, and the imaginary part is not defined. However, for all other real values $\gt 0$ the iterate is on the real axis.
Next iteration (except that point at negative infinity) gives the blod green line parallel to the real axis,with constant imaginary part of about $2.68$ Next iteration gives the parabolic blue line on the right part of the coordinate system, which is, btw, symmetric and has a nice parametric formulation.
Iterating the blue line gives the second pink line, whose two ends tend both to positive real infinity. That happens with all further iterates; interestingly they build a more and more complicated slope when the complete line is logarithmized again and again, towards the fixpoint.
To have a bit more orientation I've inserted the big red dots as reference points: the iterates of the point $(-1,0î)$:

Picture 2: I have not much more to say at the moment, and only hope that this visualizations give some initial intuition about the behave of the iteration of logarithms in the complex plane.

Iteration of logarithm

2 Answers2