TITLE: How to solve infinitely nested logarithms
QUESTION [6 upvotes]: I have an iterative process that starts with
$$x_1 = \log_{10}(a)$$
Following iterations are as follows:
$$x_2 = \log_{10}(a-b\cdot x_1)$$
$$x_3 = \log_{10}(a-b\cdot x_2)$$
$$x_4 = \log_{10}(a-b\cdot x_3)$$
$$\vdots$$
and so on indefinitely. (Assume $a$ and $b$ are known and positive)
Once these are nested, it reads:
$$x = \log_{10}(a-b\cdot \log_{10}(a-b\cdot \log_{10}(a-b \cdot \ldots \log_{10}(a-b\cdot \log_{10}(a))\ldots)))$$
This seems straight forward enough to have a simpler (non-iterative) solution, but in all my searching I can't find anything.
Any help would be appreciated.
Thanks.

REPLY [2 votes]: Take a look at the Banach fixed-point theorem. Under certain conditions this leads to the solution being:
$$x=\log_{10}(a-bx)$$
Of course, this cannot be solved analytically, but with use of the Lambert W function:
$$x = \frac{a}{b} - \frac{1}{\log 10}\cdot W\left(\frac{10^{a/b} \log 10}{b}\right)$$