User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

Theorem (Lemma?)
Let the exponential function $\exp x$ be defined as:


 * $\exp: \R \to \R_{>0}$
 * $\exp: x \mapsto \displaystyle \lim_{n \to +\infty}\left (1 + \frac x n\right)^n$

Let $e$ be Euler's Number defined as:


 * $e := \displaystyle \lim_{n \to +\infty}\left (1 + \frac 1 n\right)^n$

Then for every $x \in \N_{>0}$:


 * $\exp x = \displaystyle \prod_{j=1}^x e = \left(\underbrace{e \times e \times \dots \times e}_{x \ \text{terms}}\right) = e^x$

Log laws
Let $\ell,m,n \in \N_{>0}, x \in \R_{>0}$.


 * $x^\ell = a \iff \log_x a = \ell$


 * $x^m = b \iff \log_x b = m$


 * $x^n = a \cdot b \iff \log_x ab = n$

Corollary:


 * $\exp x = e^x, \ln x = \log_e x$

And doing this kind of thing I can prove logarithm laws for indices in $\N_{>0}$ which is enough for Derivative of Natural Logarithm Function. Thoughts? --GFauxPas 20:30, 11 January 2012 (EST)


 * My thought would be: why bother? Using this approach you are limited to natural number exponents. We've defined $e$ using the definition as given, but it doesn't make much sense to use that (raw) definition to prove stuff which can be far more easily proved using further results which follow from it.
 * As a parallel: it's like proving the derivative of a compound object by going back to first principles and ignoring the existence of the chain rule and product rule. Yeah, you can do it, but only as the sort of exercise a sadistic teacher would set as an exercise.
 * On the other hand I may be overstating the case - there could be merit in this that I can't see. Anyone else's thoughts? --prime mover 02:07, 12 January 2012 (EST)
 * I wouldn't know why this would be useful. At the start of proof 2, you use the logarithm law on $x$ and $x+\Delta x$, which mostly are not integers... Although I see a merit for a proof (if feasible) of basic results from more than one equivalent definition; although this is not strictly necessary once the equivalence of the definitions has been established as a page on PW. At the moment the work you put up does not seem like anything in that direction, hence I don't see it included on a bona fide page at this point. --Lord_Farin 04:50, 12 January 2012 (EST)
 * I think you guys are right. But hey it was good proof-writing practice :) --GFauxPas 08:14, 12 January 2012 (EST)
 * What about just the lemma, along with $\exp x = \exp \left({x \ln e}\right) = e^x$ which right now is on the definitions page? Also, LF, can you explain a little bit more about what you meant when "given uniform convergence (which is there on closed intervals), limits and differentiation may be swapped, yielding the desired immediately." --GFauxPas 08:44, 12 January 2012 (EST)

That was a statement concerning the proof of equivalence of definitions of $\exp$ (specifically, of the unique $f$ satisfying $f'=f$ and the limit definition as used on this page). To justify interchanging the limit $n\to\infty$ and differentiation, one needs some kind of uniform convergence on an interval, which is provided for the exponential function as described on WikiPedia. A uniqueness theorem from the theorem of differential equations then gives the desired equivalence of definitions. --Lord_Farin 09:52, 12 January 2012 (EST)