Discussion Overview
The discussion revolves around finding a function ##f(x)## such that ##f(f(x)) = \log_a x##, with implications for extending the definition of superlogarithms. Participants explore theoretical approaches, mathematical reasoning, and potential applications of these concepts.
Discussion Character
- Exploratory
- Technical explanation
- Conceptual clarification
- Debate/contested
- Mathematical reasoning
Main Points Raised
- Some participants propose extending the definition of superlogarithms by finding a function ##f## such that ##f(f(x)) = \log_a x##, suggesting that Taylor series might be useful.
- Others argue that defining ##\log_a x## as the number of times ##x## is divided by ##a## until reaching 1 could be problematic for values not of the form ##a^n##.
- A participant suggests that functions like ##f(x) = \frac{x}{a^{\frac{1}{2}}}## could help in this extension, proposing a method to calculate ##\log_a x## using repeated divisions.
- There is a discussion about the definition of superlogarithms and whether it should satisfy properties like ##slog_a(xy) = slog_a(x) + slog_a(y)##.
- Some participants express interest in the broader question of when a function ##g(x)## can have a corresponding function ##f## such that ##f(f(x)) = g(x)##, indicating that this might be a more interesting area of inquiry.
- Several participants engage in calculations and examples, questioning the clarity of terms like "resulting number" in the context of their proposed methods.
- A suggestion is made to use programming to numerically approximate functions that satisfy the condition ##f(f(x)) = \ln(x)##.
Areas of Agreement / Disagreement
Participants express differing views on the definitions and implications of superlogarithms and the function ##f##. There is no consensus on the best approach or definition, and the discussion remains unresolved.
Contextual Notes
Participants highlight limitations in definitions and assumptions, particularly regarding the applicability of their methods to various forms of ##x##. The discussion also reflects uncertainty about the clarity of terms used in their calculations.